%------------------------------------------------------------------------------
% File     : ITP225_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Member 00300_011794
% 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    : 0065_VEBT_Member_00300_011794 [Des22]

% Status   : Theorem
% Rating   : 0.22 v8.2.0, 0.10 v8.1.0
% Syntax   : Number of formulae    : 12621 (4600 unt;1985 typ;   0 def)
%            Number of atoms       : 23479 (9607 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 15306 (2463   ~; 528   |;1566   &)
%                                         (2639 <=>;8110  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   4 avg)
%            Maximal term depth    :   40 (   2 avg)
%            Number of types       :  321 ( 320 usr)
%            Number of type conns  : 1982 (1201   >; 781   *;   0   +;   0  <<)
%            Number of predicates  :  124 ( 121 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1544 (1544 usr; 464 con; 0-8 aty)
%            Number of variables   : 21965 (21410   !; 555   ?;21965   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 18:47:25.087
%------------------------------------------------------------------------------
% Could-be-implicit typings (320)
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J,type,
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tff(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
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tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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% Explicit typings (1665)
tff(sy_c_ATP_058Lamm__a____,type,
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tff(sy_c_ATP_058Lamm__aa____,type,
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tff(sy_c_ATP_058Lamm__af____,type,
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tff(sy_c_ATP_058Lamm__ah____,type,
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tff(sy_c_ATP_058Lamm__am____,type,
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tff(sy_c_ATP_058Lamm__an____,type,
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tff(sy_c_ATP_058Lamm__ao____,type,
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tff(sy_c_ATP_058Lamm__ap____,type,
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tff(sy_c_ATP_058Lamm__ar____,type,
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tff(sy_c_ATP_058Lamm__au____,type,
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tff(sy_c_ATP_058Lamm__av____,type,
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tff(sy_c_ATP_058Lamm__ax____,type,
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tff(sy_c_ATP_058Lamm__ay____,type,
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tff(sy_c_ATP_058Lamm__az____,type,
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    aTP_Lamm_bd: fun_int_int ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: complex > fun_na5941084329111710283x_bool ).

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

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

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_int > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_int > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_int_int > fun_int_Code_integer ).

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

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_int_int > fun_int_complex ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_nat_real ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_nat_int ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_int_int ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: nat > fun_fu8464300013002972271at_int ).

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

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_nat_complex > fun_co3258106519541737638omplex ).

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_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_complex > fun_na1142814060743658952omplex ).

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

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_int_int > fun_int_int ).

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: int > fun_complex_int ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: real > fun_int_real ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: nat > fun_complex_nat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: nat > fun_int_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: code_integer > fun_co8153928105989009288nteger ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: code_integer > fun_int_Code_integer ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: rat > fun_complex_rat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_int_int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_rat > fun_fu8328862064278946247at_rat ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: nat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_complex_nat ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_int_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_complex_nat > fun_complex_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_list_nat_nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_set_nat_nat > fun_set_nat_nat ).

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

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_complex_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_list_nat_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_set_nat_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_int_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: nat > fun_na3244735409550470327at_nat ).

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

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_re1255401033344146673l_bool ).

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

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

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_fun_nat_nat ).

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_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_int_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_real_bool ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_real_bool ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_real_real ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: nat > fun_fu3077872570260440615l_real ).

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

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

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_set_nat ).

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

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

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_rat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_int_int ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_Pr803113274115125473t_bool ).

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

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: rat > fun_Pr803113274115125473t_bool ).

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

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: nat > fun_nat_set_nat ).

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

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

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: extended_enat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: real > fun_re8658731967611918585t_real ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: nat > fun_nat_set_nat ).

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

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Oreplicate_001t__HOL__Obool,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Otake_001t__Nat__Onat,type,
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tff(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Oint__decode,type,
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tff(sy_c_Nat__Bijection_Oint__encode,type,
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tff(sy_c_Nat__Bijection_Olist__decode,type,
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tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
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tff(sy_c_Nat__Bijection_Olist__encode,type,
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tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__decode,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__encode,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Osum__decode,type,
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tff(sy_c_Nat__Bijection_Osum__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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tff(sy_c_NthRoot_Osqrt,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Onat__of__num,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onum__of__nat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Opow,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Num_Osqr,type,
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tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
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tff(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
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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,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: ( set_int * 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_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: ( set_num * set_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: ( set_rat * set_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: ( set_real * 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__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co8310303586393106575r_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Onatural,type,
    ord_le1926595141338095240atural: ( code_natural * code_natural ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: ( extended_enat * extended_enat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: ( set_complex * 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 * 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_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: ( set_rat * 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 * set_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: ( set_VEBT_VEBT * set_VEBT_VEBT ) > $o ).

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__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_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( fun_nat_fun_nat_bool * fun_nat_fun_nat_bool ) > fun_nat_bool ).

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__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).

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__Code____Numeral__Onatural,type,
    power_7079662738309270450atural: ( code_natural * nat ) > code_natural ).

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: real > fun_nat_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__Code____Numeral__Ointeger_001t__HOL__Obool,type,
    produc5135875973608410785r_bool: ( code_integer * bool ) > produc7417142470700560111r_bool ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    produc3574140220909816553atural: ( code_natural * code_natural ) > produc7822875418678951345atural ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc6639722614265839536atural: code_natural > fun_Pr7769407664385210434atural ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
    produc8297744024737662807l_bool: ( bool * bool ) > produc2804947525362035943l_bool ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
    produc8871513391732212460ol_int: ( bool * int ) > produc5612625445546783218ol_int ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
    produc8874003862241262736ol_nat: ( bool * nat ) > produc567104428201204118ol_nat ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    produc9074556814250467700T_VEBT: ( bool * vEBT_VEBT ) > produc8408751633324799108T_VEBT ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: ( product_prod_nat_nat * product_prod_nat_nat ) > produc859450856879609959at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: ( vEBT_VEBT * extended_enat ) > produc7272778201969148633d_enat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc3035007008238225530T_bool: ( vEBT_VEBT * bool ) > produc6448083994118892042T_bool ).

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

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: ( set_nat * fun_nat_set_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( fun_Co3583066429914935172er_nat * produc8923325533196201883nteger ) > nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( fun_Co8521658056209057742er_num * produc8923325533196201883nteger ) > num ).

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__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    produc6117575242473779287r_bool: ( fun_Co3025358997185726883r_bool * produc8923325533196201883nteger ) > produc7417142470700560111r_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
    produc8560705615207772305t_bool: fun_int_fun_int_bool > fun_Pr803113274115125473t_bool ).

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__HOL__Obool,type,
    produc8955392012077580105t_bool: fun_nat_fun_nat_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( fun_na819858167508000762st_nat * product_prod_nat_nat ) > list_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: fun_nat_fun_nat_nat > fun_Pr2602117396471556196at_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 > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
    produc5023667602222594023t_bool: fun_na4786964484885125043t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4528391901023682385at_nat: fun_na1668100129766719005at_nat > fun_Pr2477239776880934807at_nat ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oscomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc5538323210962509403atural: ( fun_Pr7769407664385210434atural * fun_Co1105491363547889651atural ) > fun_Pr7769407664385210434atural ).

tff(sy_c_Quotient_OQuotient3_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    quotie6776551016481293500at_int: ( fun_Pr248683899324676269t_bool * fun_Pr7647638413817135296at_int * fun_in4514472496968388910at_nat ) > $o ).

tff(sy_c_Quotient_OQuotient3_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    quotie8981163687542601860t_real: ( fun_fu2636726645394866687t_bool * fun_fun_nat_rat_real * fun_real_fun_nat_rat ) > $o ).

tff(sy_c_Random_Oinc__shift,type,
    inc_shift: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Oiterate_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    iterat8892046348760725948atural: ( code_natural * fun_Co1105491363547889651atural * code_natural ) > fun_Pr7769407664385210434atural ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun_Pr4637083014979798189l_bool ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun_Pr7769407664385210434atural ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun_Pr7769407664385210434atural ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun_Pr1652463973008025892nt_rat ).

tff(sy_c_Rat_OFract,type,
    fract: fun_int_fun_int_rat ).

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

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: fun_rat_real ).

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

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr3145601080792579042t_bool ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_bool ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Real_ORatreal,type,
    ratreal: fun_rat_real ).

tff(sy_c_Real_OReal,type,
    real2: fun_fun_nat_rat_real ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun_nat_rat > $o ).

tff(sy_c_Real_Ocr__real,type,
    cr_real: fun_fu990398353789569848l_bool ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu990398353789569848l_bool ).

tff(sy_c_Real_Opositive,type,
    positive2: fun_real_bool ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu2636726645394866687t_bool ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun_real_fun_nat_rat ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun_nat_rat > bool ).

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: real > fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_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: fun_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,
    real_V2046097035970521341omplex: real > fun_complex_complex ).

tff(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Relation_Otransp_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    transp_fun_nat_rat: fun_fu2636726645394866687t_bool > $o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: fun_nat_fun_nat_bool ).

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__Code____Numeral__Onatural,type,
    divide5121882707175180666atural: ( code_natural * code_natural ) > code_natural ).

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: 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: 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_Co8310303586393106575r_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
    dvd_dvd_Code_natural: ( code_natural * code_natural ) > $o ).

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

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_re1255401033344146673l_bool ).

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__Code____Numeral__Onatural,type,
    modulo8411746178871703098atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: 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_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
    unit_f2748546683901255202or_nat: fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: bool > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
    zero_n8403883297036319079atural: bool > code_natural ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: bool > complex ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_bool_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: fun_bool_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: bool > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: bool > real ).

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__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_bool ).

tff(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: fun_nat_int > fun_int_bool ).

tff(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: fun_nat_nat > fun_nat_bool ).

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

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

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

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_bool > set_list_nat ).

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

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr803113274115125473t_bool > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: fun_Pr7757163693017731481t_bool > set_Pr1261947904930325089at_nat ).

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

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

tff(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

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__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_list_nat_nat: ( fun_list_nat_nat * set_list_nat ) > 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__List__Olist_It__Nat__Onat_J,type,
    image_nat_list_nat: ( fun_nat_list_nat * set_nat ) > set_list_nat ).

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__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_5846123807819985514at_nat: ( fun_na1476630412504255114at_nat * set_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_678696785212003926at_nat: ( fun_na3635074868517587788at_nat * set_nat ) > set_Sum_sum_nat_nat ).

tff(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_2486076414777270412at_nat: ( fun_Pr2602117396471556196at_nat * set_Pr1261947904930325089at_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__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_1320371278474632150at_nat: ( fun_Su884838341992399436at_nat * set_Sum_sum_nat_nat ) > set_nat ).

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__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: ( list_nat * set_list_nat ) > set_list_nat ).

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

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

tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: ( set_nat * set_set_nat ) > set_set_nat ).

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_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( fun_na7078387940644931716nteger * nat * nat * code_integer ) > code_integer ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat * nat ) > fun_complex_complex ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( fun_nat_fun_int_int * nat * nat ) > fun_int_int ).

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_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat ) > fun_rat_rat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( fun_na6724076748692969250l_real * nat ) > fun_na6724076748692969250l_real ).

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__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_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

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_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: 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__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Num__Onum,type,
    set_or2392100141987894638an_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Rat__Orat,type,
    set_or5199638295745620268an_rat: ( rat * rat ) > set_rat ).

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_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or8625682525731655386et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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_String_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: ( literal * fun_Pr1792556611952799308t_real ) > real ).

tff(sy_c_String_OLiteral,type,
    literal2: ( bool * bool * bool * bool * bool * bool * bool * literal ) > literal ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inl_nat_nat: nat > sum_sum_nat_nat ).

tff(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inr_nat_nat: nat > sum_sum_nat_nat ).

tff(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    sum_ca7763040182479039464nt_nat: ( fun_nat_int * fun_nat_int * sum_sum_nat_nat ) > int ).

tff(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ca6763686470577984908at_nat: ( fun_nat_nat * fun_nat_nat * sum_sum_nat_nat ) > 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_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: ( set_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_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: fun_nat_real > $o ).

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__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

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: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

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: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log2: 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_Opowr__real,type,
    powr_real2: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr5442483421567724333t_bool ).

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_bool ).

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_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

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

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_bool ).

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_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).

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_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li8598321315282354825t_bool > fun_list_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    accp_P8126237942716283194atural: ( fun_Pr4637083014979798189l_bool * produc7822875418678951345atural ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( fun_Pr3099820731930151597t_bool * product_prod_int_int ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( fun_Pr248683899324676269t_bool * product_prod_nat_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr817965127378462947t_bool * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE5877677516038742509T_bool > fun_VEBT_VEBT_bool ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
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    aa_com4217951221296830281nteger: ( fun_co8153928105989009288nteger * complex ) > code_integer ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Int__Oint,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    aa_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_aa_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Int__Oint_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Rat__Orat,type,
    aa_int_rat: ( fun_int_rat * int ) > rat ).

tff(sy_c_aa_001t__Int__Oint_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__List__Olist_It__HOL__Obool_J_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__List__Olist_It__Int__Oint_J_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Num__Onum,type,
    aa_nat_num: ( fun_nat_num * nat ) > num ).

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tff(sy_c_aa_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    aa_nat_set_nat: ( fun_nat_set_nat * nat ) > set_nat ).

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tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_bool ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: ( real * set_real ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10569)
tff(fact_0_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_2_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_3_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_4__C3_Ohyps_C_I3_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "3.hyps"(3)
tff(fact_5_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_6_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_7_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_8_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_9_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_10_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_11_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_12_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_13_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_14_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_15_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_16_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_17_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_18_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_19_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_20_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M2: nat] : N = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_21_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_22_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_23_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_24_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_25_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X3),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_26_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_27_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_28_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_29_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_30_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_31_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_32_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_33_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => pp(aa_nat_bool(P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
               => pp(aa_nat_bool(P,I2)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_34_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_35_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_36_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_37_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_38_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M3: nat] :
          ( ( M = aa_nat_nat(suc,M3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M3)) ) ) ).

% Suc_less_eq2
tff(fact_39_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
           => pp(aa_nat_bool(P,I3)) ) ) ) ).

% All_less_Suc
tff(fact_40_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_41_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_42_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_bool] :
      ( member_complex(A,collect_complex(P))
    <=> pp(aa_complex_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_43_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( member_real(A,collect_real(P))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_46_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_47_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( member_int(A,collect_int(P))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_48_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_49_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_50_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_51_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_52_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_53_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_54_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(P,X3))
        <=> pp(aa_real_bool(Q,X3)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_55_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X3: list_nat] :
          ( pp(aa_list_nat_bool(P,X3))
        <=> pp(aa_list_nat_bool(Q,X3)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_56_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X3: set_nat] :
          ( pp(aa_set_nat_bool(P,X3))
        <=> pp(aa_set_nat_bool(Q,X3)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_57_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_nat_bool(P,X3))
        <=> pp(aa_nat_bool(Q,X3)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_58_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_int_bool(P,X3))
        <=> pp(aa_int_bool(Q,X3)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_59_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
            & pp(aa_nat_bool(P,I3)) ) ) ) ).

% Ex_less_Suc
tff(fact_60_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_61_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_62_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_63_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_64_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_65_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_66_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_67_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_68_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_69_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_70_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_71_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_72_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_73_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_74_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_75_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_76_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_77_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M2: nat] : N = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_78_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% All_less_Suc2
tff(fact_79_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% gr0_conv_Suc
tff(fact_80_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% Ex_less_Suc2
tff(fact_81_zero__reorient,axiom,
    ! [X: literal] :
      ( ( zero_zero_literal = X )
    <=> ( X = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_82_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_83_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_84_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_85_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_86_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_87_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
              & ~ pp(aa_nat_bool(P,M5)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_88_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
             => pp(aa_nat_bool(P,M5)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_89_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_90_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_91_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_92_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_93_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_94_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_95_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_96_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_97_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_98_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                  & ~ pp(aa_nat_bool(P,M5)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_99__C3_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "3.hyps"(4)
tff(fact_100_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_101_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_102_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_103_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_104_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D2))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_105_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_106_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_107_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_108_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_109__C3_Ohyps_C_I1_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "3.hyps"(1)
tff(fact_110_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_111_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_112_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_113_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_114_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_115_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_116_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_117_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_118_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_119_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_129_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_130_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_131_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_132_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_133_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_158_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_159_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_160_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_161_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_162_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_163_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_164_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_165_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_166_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_167_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_168_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_169_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_170_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_171_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_172_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_173_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_174_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_175_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_176_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_177_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_178_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_179_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_180_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_181_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_182_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_183_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_184_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_185_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_186_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_187_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_188_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_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_189_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_190_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_191_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_192_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_193_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_194_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_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(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_195_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_196_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_197_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_198_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_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_199_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_200_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_201_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_202_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_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(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_203_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_204_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_205_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_206_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_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_207_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_208_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_209_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_210_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_211_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_212_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_213_Suc__diff__diff,axiom,
    ! [M: nat,N: 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)),N)),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),N)),K) ).

% Suc_diff_diff
tff(fact_214_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_215_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_216_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_217_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_218_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_219_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_220_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_221_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_222_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_223_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_224_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_225_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_226_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel1
tff(fact_227_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel1
tff(fact_228_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel1
tff(fact_229_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel1
tff(fact_230_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel2
tff(fact_231_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel2
tff(fact_232_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel2
tff(fact_233_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel2
tff(fact_234_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_235_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_236_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_237_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_238_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_239_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_240_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_241_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_242_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_243_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_244_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_245_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_246_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_247_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_248_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_249_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_250_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_251_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_252_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_253_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_254_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_255_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_256_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_257_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_258_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_259_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_260_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_261_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_262_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_263_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_264_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_265_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_266_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_267_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_add
tff(fact_268_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_269_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_270_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_271__C3_Ohyps_C_I5_J,axiom,
    ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),X_1)) ).

% "3.hyps"(5)
tff(fact_272_ab__semigroup__add__class_Oadd__ac_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)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_273_ab__semigroup__add__class_Oadd__ac_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)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_274_ab__semigroup__add__class_Oadd__ac_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,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)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_275_ab__semigroup__add__class_Oadd__ac_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)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_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(minus_minus_rat(A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_290_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_291_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_292_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_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_293_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_294_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_295_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_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_296_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_297_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_298_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(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).

% add_diff_eq
tff(fact_299_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_300_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_301_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_diff_eq2
tff(fact_302_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_303_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_304_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_305_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_306_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_307_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_308_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_add_eq
tff(fact_309_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_310_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_311_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_312_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_313_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_314_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_315_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_316_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_317_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_318_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_319_add_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) ).

% add.commute
tff(fact_320_add_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) ).

% add.commute
tff(fact_321_add_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) ).

% add.commute
tff(fact_322_add_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) ).

% add.commute
tff(fact_323_add_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)) ).

% add.left_commute
tff(fact_324_add_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)) ).

% add.left_commute
tff(fact_325_add_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)) ).

% add.left_commute
tff(fact_326_add_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)) ).

% add.left_commute
tff(fact_327_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_328_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_329_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_330_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_331_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_332_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_333_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_334_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_335_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_336_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_337_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_338_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_339_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(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_340_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_341_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_342_diff__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) ).

% diff_right_commute
tff(fact_343_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_344_diff__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) ).

% diff_right_commute
tff(fact_345_diff__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) ).

% diff_right_commute
tff(fact_346_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: 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),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_347_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: 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,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_348_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_349_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_350_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_351_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_352_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_353_diff__add__inverse,axiom,
    ! [N: 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,N),M)),N) = M ).

% diff_add_inverse
tff(fact_354_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_355_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_356_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_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(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_357_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_358_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_359_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_360_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_361_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(minus_minus_real(C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% less_diff_eq
tff(fact_362_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).

% less_diff_eq
tff(fact_363_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% less_diff_eq
tff(fact_364_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_less_eq
tff(fact_365_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% diff_less_eq
tff(fact_366_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_less_eq
tff(fact_367_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_368_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_369_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_370_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_371_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% reals_Archimedean2
tff(fact_372_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D3: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
           => pp(aa_nat_bool(P,D3)) ) ) ) ).

% nat_diff_split
tff(fact_373_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D3: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
              & ~ pp(aa_nat_bool(P,D3)) ) ) ) ).

% nat_diff_split_asm
tff(fact_374_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_375_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_376_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_377_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D))) ) ) ).

% diff_strict_mono
tff(fact_378_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_strict_mono
tff(fact_379_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_380_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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_381_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_382_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) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_383_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_384_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_strict_left_mono
tff(fact_385_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_386_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_387_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_strict_right_mono
tff(fact_388_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_401_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_402_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_403_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_404_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_405_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_406_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_407_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_408_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_409_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_410_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_411_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_412_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_413_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_414_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_415_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_416_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_417_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_418_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_419_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_420_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_421_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_422_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_423_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_424_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_425_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_426_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_427_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_428_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_429_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_430_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_431_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_432_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_433_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_434_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_435_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_436_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_437_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_438_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_439_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_440_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_441_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_442_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_443_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_444_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_445_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_446_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_447_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_448_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_449_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_450_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_451_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_452_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_453_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)) ).

% of_nat_less_0_iff
tff(fact_454_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_455_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_456_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_457_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_458_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_459_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_460_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_461_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_462_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_463_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ) ).

% less_imp_of_nat_less
tff(fact_464_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_465_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_466_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_467_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_468_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_469_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_470_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_471_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_472_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_473_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_474_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_475_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_476_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_477_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_pos
tff(fact_478_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_pos
tff(fact_479_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_pos
tff(fact_480_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_pos
tff(fact_481_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_482_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% pos_add_strict
tff(fact_483_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% pos_add_strict
tff(fact_484_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% pos_add_strict
tff(fact_485_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% pos_add_strict
tff(fact_486_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_487_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_488_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_489_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_490_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_491_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_492_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_493_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_494_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_495_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q2: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ) ).

% less_natE
tff(fact_496_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_497_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_498_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_499_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_500_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_501_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_502_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_503_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_504_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_505_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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_506_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_507_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_508_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_509_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq2
tff(fact_510_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).

% valid_eq1
tff(fact_511_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq
tff(fact_512_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_513_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_514_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_515_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_516_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_517_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_518_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_519_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_520_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_521_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_522_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_523_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_524_zadd__int__left,axiom,
    ! [M: nat,N: 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,N)),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),N))),Z) ).

% zadd_int_left
tff(fact_525_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_526_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_527_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_528_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12)) ) ).

% not_min_Null_member
tff(fact_529__C3_OIH_C_I2_J,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),X))
     => pp(aa_nat_bool(vEBT_vebt_member(summary),X)) ) ).

% "3.IH"(2)
tff(fact_530_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_531_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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(minus_minus_nat,A),B)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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(minus_minus_nat,A),B)) = 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_532_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(2)
tff(fact_533_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_534_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_535_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_536_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_546_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_547_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_548_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_549_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_550_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_551_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_552_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_553_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_554_int__if,axiom,
    ! [P: bool,A: nat,B: nat] :
      ( ( pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,A) ) )
      & ( ~ pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ) ).

% int_if
tff(fact_555__C3_OIH_C_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X4),na))
        & ! [Xa: nat] :
            ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),Xa))
           => pp(aa_nat_bool(vEBT_vebt_member(X4),Xa)) ) ) ) ).

% "3.IH"(1)
tff(fact_556_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_557_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_558_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_559_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_560_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),X)) ) ) ).

% ex_inverse_of_nat_less
tff(fact_561_ex__inverse__of__nat__less,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),X)) ) ) ).

% ex_inverse_of_nat_less
tff(fact_562_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_563_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_564_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_565_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_566_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_567_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_568_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_569_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_570_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_571_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% zero_less_ceiling
tff(fact_572_zero__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_ceiling
tff(fact_573_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_574_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_575__C3_Ohyps_C_I6_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) ).

% "3.hyps"(6)
tff(fact_576_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ).

% deg1Leaf
tff(fact_577_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ).

% deg_1_Leaf
tff(fact_578_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_579_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_580_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_581_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_582_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_583_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_584_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_585_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_586_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_587_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_588_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_589_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_590_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_591_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_592_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_593_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_594_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_595_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,W) = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_596_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_597_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,W) = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_598_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_599_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X22: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X22) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X22)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_600_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_601_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_602_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_603_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_604_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_605_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_606_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_607_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_608_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_609_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_610_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_611_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_612_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_613_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_614_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_615_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_616_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_617_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_618_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_619_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_620_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_621_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_622_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_623_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% neg_less_iff_less
tff(fact_624_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% neg_less_iff_less
tff(fact_625_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_626_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_627_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_628_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_629_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_630_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_631_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_632_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_633_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_634_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_635_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = B ).

% minus_add_cancel
tff(fact_636_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_637_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_638_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_639_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_640_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) = B ).

% add_minus_cancel
tff(fact_641_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_642_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_643_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_644_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_645_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A) ).

% minus_diff_eq
tff(fact_646_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_647_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_648_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_649_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = one_one_Code_integer )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_650_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_651_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_652_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_653_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_654_of__int__1,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,one_one_int) = one_one_Code_integer ).

% of_int_1
tff(fact_655_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_656_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_657_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_658_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_659_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_660_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_661_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_662_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)) = X )
    <=> ? [N4: int] : X = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_ceiling_cancel
tff(fact_663_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)) = X )
    <=> ? [N4: int] : X = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_ceiling_cancel
tff(fact_664_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_665_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_666_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_667_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_0_iff_less
tff(fact_668_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_669_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_670_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_671_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_672_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_673_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_674_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_675_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_pos
tff(fact_676_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_677_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_678_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_679_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_680_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_681_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_682_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_683_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_684_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_685_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_686_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_687_add_Oright__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_688_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_689_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_690_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_691_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_692_ab__left__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_693_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_694_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_695_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_696_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_697_diff__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_698_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_699_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_700_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_701_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_702_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_703_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_704_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_705_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_706_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_707_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A) ).

% uminus_add_conv_diff
tff(fact_708_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(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_709_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_710_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_711_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_712_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) ).

% diff_minus_eq_add
tff(fact_713_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_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_714_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_715_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_716_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = zero_z3403309356797280102nteger )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_717_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_718_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_719_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_720_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_721_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_z3403309356797280102nteger = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_722_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_723_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_724_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_725_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_726_of__int__0,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,zero_zero_int) = zero_z3403309356797280102nteger ).

% of_int_0
tff(fact_727_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_728_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_729_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_730_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_731_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_732_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_733_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_734_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_735_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_736_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_737_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_738_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_739_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_740_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_741_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_742_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_743_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_744_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_745_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_746_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_747_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_748_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_749_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_750_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_751_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_752_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_753_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_754_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_755_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_add
tff(fact_756_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_757_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_complex_complex(plus_plus_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_add
tff(fact_758_of__int__minus,axiom,
    ! [Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_minus
tff(fact_759_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_760_of__int__minus,axiom,
    ! [Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,Z)) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_minus
tff(fact_761_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_762_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_763_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_764_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_765_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_diff
tff(fact_766_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_complex_complex(minus_minus_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_diff
tff(fact_767_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_768_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(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_769_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_770_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_771_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ).

% of_int_of_nat_eq
tff(fact_772_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% of_int_of_nat_eq
tff(fact_773_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% of_int_of_nat_eq
tff(fact_774_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% of_int_of_nat_eq
tff(fact_775_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% of_int_of_nat_eq
tff(fact_776_ceiling__of__nat,axiom,
    ! [N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% ceiling_of_nat
tff(fact_777_ceiling__of__nat,axiom,
    ! [N: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% ceiling_of_nat
tff(fact_778_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_779_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_780_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_781_add__neg__numeral__special_I7_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_782_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_783_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_784_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_785_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_786_add__neg__numeral__special_I8_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_787_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_788_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_789_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_790_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_791_diff__numeral__special_I12_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_792_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_793_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_794_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_795_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_796_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_797_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_798_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_799_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_800_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_801_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_802_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_803_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_804_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_805_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_806_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_807_one__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).

% one_less_ceiling
tff(fact_808_one__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).

% one_less_ceiling
tff(fact_809_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% zless_nat_conj
tff(fact_810_ceiling__add__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_add_one
tff(fact_811_ceiling__add__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_812_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_813_ceiling__diff__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_diff_one
tff(fact_814_ceiling__diff__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_diff_one
tff(fact_815_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_816_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),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,X)),Z) ).

% ceiling_add_of_int
tff(fact_817_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X)),Z) ).

% ceiling_add_of_int
tff(fact_818_ceiling__diff__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),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,X)),Z) ).

% ceiling_diff_of_int
tff(fact_819_ceiling__diff__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(X),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,X)),Z) ).

% ceiling_diff_of_int
tff(fact_820_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_821_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_822_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_823_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_824_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_825_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_826_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_827_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_828_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_829_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X222: bool] : Y != vEBT_Leaf(X212,X222) ) ).

% VEBT.exhaust
tff(fact_830_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X22: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X22) ).

% VEBT.distinct(1)
tff(fact_831_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_832_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_833_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_834_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_835_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_836_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_837_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_838_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_839_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_840_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_841_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_842_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_843_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_844_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_845_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_846_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_847_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_848_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_849_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_850_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_851_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_852_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_853_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_854_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_855_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_856_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_857_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_858_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_859_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_860_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_861_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_int(ring_1_of_int_int,K) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_862_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_real(ring_1_of_int_real,K) = aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_863_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_864_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_rat(ring_1_of_int_rat,K) = aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_865_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_866_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_867_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_868_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_869_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_870_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_871_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_872_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_873_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_874_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_875_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X)) ).

% ex_of_int_less
tff(fact_876_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)) ).

% ex_of_int_less
tff(fact_877_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_less_of_int
tff(fact_878_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_less_of_int
tff(fact_879_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_880_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_881_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_882_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E2)) ) ) ).

% real_arch_inverse
tff(fact_883_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_884_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_less_iff
tff(fact_885_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_less_iff
tff(fact_886_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_less_iff
tff(fact_887_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_less_iff
tff(fact_888_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_889_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_890_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% less_minus_iff
tff(fact_891_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% less_minus_iff
tff(fact_892_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_893_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_894_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_895_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_896_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_897_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_898_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_899_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_900_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_901_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_902_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_903_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_904_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_905_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_906_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_907_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_908_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_909_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_910_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_911_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_912_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_913_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_914_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ).

% minus_diff_minus
tff(fact_915_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_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(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_916_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_917_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_918_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_919_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ).

% minus_diff_commute
tff(fact_920_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_921_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_922_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_923_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_924_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_925_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_926_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_927_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_928_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_929_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_930_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_931_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% int_cases2
tff(fact_932_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_933_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_934_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_935_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_936_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_937_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_938_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_939_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_940_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_941_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_942_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_943_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_944_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_945_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_946_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_947_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_948_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_949_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_950_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_951_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_952_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_953_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_954_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_955_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_956_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_957_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_958_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_959_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_960_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_961_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_962_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_963_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_964_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_965_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_966_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_967_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_968_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_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_969_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_970_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_971_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_972_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_973_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_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_974_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_975_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_976_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_977_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_978_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(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(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_979_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_980_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_981_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_982_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_983_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_984_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_985_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_986_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_987_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_988_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_989_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_990_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_991_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_992_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z: int] :
      ( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
       => pp(aa_int_bool(P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_993_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_994_add__mono1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_995_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_996_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_997_add__mono1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_998_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).

% less_add_one
tff(fact_999_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat))) ).

% less_add_one
tff(fact_1000_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_1001_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_1002_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_1003_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% not_int_zless_negative
tff(fact_1004_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1005_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1006_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1007_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1008_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_1009_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1010_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_1011_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_1012_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_1013_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_1014_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_1015_int__minus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),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,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_1016_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1017_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1018_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1019_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(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_1020_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_1021_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1022_reals__Archimedean,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),X)) ) ).

% reals_Archimedean
tff(fact_1023_reals__Archimedean,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),X)) ) ).

% reals_Archimedean
tff(fact_1024_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))) ) ).

% of_int_pos
tff(fact_1025_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_pos
tff(fact_1026_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_pos
tff(fact_1027_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_pos
tff(fact_1028_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_1029_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_1030_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_1031_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_1032_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N4: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => pp(aa_nat_bool(P,N4)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_1033_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_1034_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_1035_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_1036_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1037_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1038__C3_Oprems_C,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(none_P5556105721700978146at_nat,deg,treeList,summary)),xa)) ).

% "3.prems"
tff(fact_1039_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% inverse_positive_iff_positive
tff(fact_1040_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% inverse_positive_iff_positive
tff(fact_1041_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% inverse_negative_iff_negative
tff(fact_1042_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% inverse_negative_iff_negative
tff(fact_1043_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_1044_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_1045_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_1046_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_1047_inverse__minus__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ).

% inverse_minus_eq
tff(fact_1048_inverse__minus__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ).

% inverse_minus_eq
tff(fact_1049_inverse__minus__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ).

% inverse_minus_eq
tff(fact_1050_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(inverse_inverse_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_1051_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,X) = one_one_complex )
    <=> ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_1052_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,X) = one_one_rat )
    <=> ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_1053_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_1054_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_1055_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_1056_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_1057_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_1058_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_1059_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_1060_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_1061_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_1062_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X = A ) ) ).

% real_add_minus_iff
tff(fact_1063_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_1064_inverse__inverse__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ).

% inverse_inverse_eq
tff(fact_1065_inverse__inverse__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ).

% inverse_inverse_eq
tff(fact_1066_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_1067_inverse__eq__iff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_1068_inverse__eq__iff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_1069_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_1070_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_1071_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_1072_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_1073_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_1074_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1075_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_1076_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_1077_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_1078_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_1079_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_1080_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1081_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X4)) ).

% linordered_field_no_lb
tff(fact_1082_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X4)) ).

% linordered_field_no_lb
tff(fact_1083_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X_12)) ).

% linordered_field_no_ub
tff(fact_1084_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),X_12)) ).

% linordered_field_no_ub
tff(fact_1085_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1086_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_1087_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_1088_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_1089_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_1090_inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_1091_inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_1092_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1093_Suc__as__int,axiom,
    ! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),one_one_int)) ).

% Suc_as_int
tff(fact_1094_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_1095_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_1096_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_1097_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_1098_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_1099_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_1100_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_1101_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_1102_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_1103_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_1104_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_1105_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_1106_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_1107_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_1108_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_1109_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% inverse_less_imp_less
tff(fact_1110_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% inverse_less_imp_less
tff(fact_1111_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% less_imp_inverse_less
tff(fact_1112_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% less_imp_inverse_less
tff(fact_1113_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% inverse_less_imp_less_neg
tff(fact_1114_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% inverse_less_imp_less_neg
tff(fact_1115_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_1116_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_1117_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
     => ( ( A != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% inverse_negative_imp_negative
tff(fact_1118_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
     => ( ( A != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% inverse_negative_imp_negative
tff(fact_1119_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
     => ( ( A != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ).

% inverse_positive_imp_positive
tff(fact_1120_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
     => ( ( A != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ).

% inverse_positive_imp_positive
tff(fact_1121_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)) ) ).

% negative_imp_inverse_negative
tff(fact_1122_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)) ) ).

% negative_imp_inverse_negative
tff(fact_1123_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))) ) ).

% positive_imp_inverse_positive
tff(fact_1124_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ).

% positive_imp_inverse_positive
tff(fact_1125_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_1126_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_1127_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_1128_one__less__inverse,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A))) ) ) ).

% one_less_inverse
tff(fact_1129_one__less__inverse,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% one_less_inverse
tff(fact_1130_one__less__inverse__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,X)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% one_less_inverse_iff
tff(fact_1131_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% one_less_inverse_iff
tff(fact_1132_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_1133_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_1134_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_1135_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_1136_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_1137_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_1138_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,N)),one_one_rat)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_1139_ceiling__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).

% ceiling_less_zero
tff(fact_1140_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ) ).

% ceiling_less_zero
tff(fact_1141_zero__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).

% zero_le_ceiling
tff(fact_1142_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)) ) ).

% zero_le_ceiling
tff(fact_1143_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_1144_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_1145_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_1146_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_1147_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_1148_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_1149_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_1150_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_1151_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_1152_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1153_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1154_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_1155_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1156_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1157_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_1158_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_1159_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1160_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1161_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_1162_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% neg_le_iff_le
tff(fact_1163_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_1164_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% neg_le_iff_le
tff(fact_1165_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% neg_le_iff_le
tff(fact_1166_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_1167_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1168_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1169_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1170_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1171_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_1172_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_1173_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1174_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1175_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1176_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1177_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_1178_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_1179_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_1180_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_1181_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_1182_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_1183_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_1184_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_1185_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_1186_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_1187_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_1188_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_1189_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_1190_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel1
tff(fact_1191_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel1
tff(fact_1192_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel1
tff(fact_1193_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel1
tff(fact_1194_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel2
tff(fact_1195_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel2
tff(fact_1196_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel2
tff(fact_1197_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel2
tff(fact_1198_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1199_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1200_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1201_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1202_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1203_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1204_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_1205_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_1206_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_1207_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_1208_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_1209_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_1210_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_1211_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_1212_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_1213_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_le_0_iff_le
tff(fact_1214_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_1215_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_1216_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_1217_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_1218_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_1219_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_1220_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_1221_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_less_eq_nonneg
tff(fact_1222_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_1223_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1224_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1225_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1226_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1227_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1228_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1229_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1230_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1231_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_1232_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_1233_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_1234_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_1235_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_1236_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_1237_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_1238_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_1239_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1240_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1241_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_1242_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_1243_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_1244_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1245_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1246_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1247_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1248_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1249_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1250_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_1251_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_1252_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_1253_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% inverse_le_iff_le
tff(fact_1254_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% inverse_le_iff_le
tff(fact_1255_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_1256_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_1257_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% zle_add1_eq_le
tff(fact_1258_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_1259_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% zle_diff1_eq
tff(fact_1260_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_1261_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_1262_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_1263_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_1264_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_1265_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_1266_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_1267_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_1268_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_1269_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_1270_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_1271_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_1272_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_1273_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_1274_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_1275_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_1276_ceiling__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_1277_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_le_zero
tff(fact_1278_ceiling__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_1279_ceiling__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_less_one
tff(fact_1280_one__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% one_le_ceiling
tff(fact_1281_one__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_le_ceiling
tff(fact_1282_ceiling__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% ceiling_le_one
tff(fact_1283_ceiling__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).

% ceiling_le_one
tff(fact_1284_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,Z)) = aa_int_complex(ring_17405671764205052669omplex,Z) ) ) ).

% of_nat_nat
tff(fact_1285_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_1286_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1287_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1288_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1289_minus__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(minus_minus_real(X),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_1290_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% verit_la_disequality
tff(fact_1291_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% verit_la_disequality
tff(fact_1292_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% verit_la_disequality
tff(fact_1293_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% verit_la_disequality
tff(fact_1294_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
      | ( A = X )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A)) ) ).

% verit_la_generic
tff(fact_1295_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1296_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1297_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1298_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1299_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1300_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_1301_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_1302_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_1303_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_1304_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_1305_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1306_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1307_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1308_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1309_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1310_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_1311_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J))) ) ).

% of_nat_mono
tff(fact_1312_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_1313_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_1314_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_1315_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_1316_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(3)
tff(fact_1317_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_1318_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_mono
tff(fact_1319_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_mono
tff(fact_1320_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_1321_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% ceiling_le_iff
tff(fact_1322_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% ceiling_le_iff
tff(fact_1323_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A)) ) ).

% ceiling_le
tff(fact_1324_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),A)) ) ).

% ceiling_le
tff(fact_1325_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_1326_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_1327_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_1328_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_1329_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_1330_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B5),A5))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1331_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B5),A5))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1332_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B5),A5))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1333_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B5),A5))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1334_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),A5))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1335_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_1336_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_1337_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_1338_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_1339_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1340_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1341_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1342_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1343_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1344_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1345_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1346_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1347_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_1348_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1349_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1350_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1351_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1352_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_1353_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1354_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1355_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1356_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1357_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1358_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1359_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1360_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1361_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1362_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1363_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1364_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1365_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1366_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1367_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1368_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1369_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1370_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1371_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1372_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1373_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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1374_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1375_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) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1376_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1377_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_right_mono
tff(fact_1378_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1379_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1380_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_left_mono
tff(fact_1381_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1382_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D))) ) ) ).

% diff_mono
tff(fact_1383_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_mono
tff(fact_1384_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1385_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_1386_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_imp_neg_le
tff(fact_1387_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_imp_neg_le
tff(fact_1388_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_1389_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_le_iff
tff(fact_1390_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_le_iff
tff(fact_1391_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_le_iff
tff(fact_1392_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_le_iff
tff(fact_1393_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_1394_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_minus_iff
tff(fact_1395_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_minus_iff
tff(fact_1396_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_1397_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).

% compl_mono
tff(fact_1398_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X)))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).

% compl_le_swap1
tff(fact_1399_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y)) ) ).

% compl_le_swap2
tff(fact_1400_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_1401_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_1402_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_1403_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M6))
     => ? [M2: nat] : M6 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_1404_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_1405_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_1406_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_1407_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M5)),N2))
             => pp(aa_nat_bool(P,M5)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_1408_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_1409_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X3),X3))
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X3),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X3),Z2)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1410_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1411_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1412_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1413_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_1414_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),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_1415_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_1416_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% real_arch_simple
tff(fact_1417_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_le_0_iff
tff(fact_1418_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_1419_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_1420_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_1421_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_1422_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_1423_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_1424_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_1425_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_1426_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_1427_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_1428_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_1429_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_1430_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1431_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1432_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_1433_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_1434_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_1435_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% le_add2
tff(fact_1436_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

% le_add1
tff(fact_1437_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_1438_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_1439_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_1440_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

% diff_le_self
tff(fact_1441_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),L))) ) ).

% diff_le_mono
tff(fact_1442_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( 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,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1443_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_1444_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1445_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_le_of_int
tff(fact_1446_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_le_of_int
tff(fact_1447_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).

% int_less_real_le
tff(fact_1448_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_nonneg
tff(fact_1449_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))) ) ).

% of_int_nonneg
tff(fact_1450_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_nonneg
tff(fact_1451_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_nonneg
tff(fact_1452_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_1453_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_1454_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_1455_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_1456_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_1457_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_1458_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1459_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1460_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1461_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1462_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_1463_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1464_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1465_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_1466_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1467_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1468_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1469_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1470_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing
tff(fact_1471_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_increasing
tff(fact_1472_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing
tff(fact_1473_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing
tff(fact_1474_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1475_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1476_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1477_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1478_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing2
tff(fact_1479_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_increasing2
tff(fact_1480_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing2
tff(fact_1481_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing2
tff(fact_1482_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1483_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1484_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1485_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1486_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_1487_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1488_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1489_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1490_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1491_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1492_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1493_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1494_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1495_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1496_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1497_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1498_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_1499_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_1500_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_1501_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1502_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1503_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1504_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1505_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1506_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1507_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1508_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1509_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1510_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1511_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1512_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1513_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1514_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1515_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1516_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1517_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(minus_minus_real(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1518_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1519_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1520_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_1521_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1522_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1523_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1524_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1525_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1526_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1527_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1528_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1529_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1530_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1531_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1532_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1533_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1534_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) ) ).

% diff_add
tff(fact_1535_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% le_diff_eq
tff(fact_1536_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).

% le_diff_eq
tff(fact_1537_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% le_diff_eq
tff(fact_1538_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_le_eq
tff(fact_1539_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% diff_le_eq
tff(fact_1540_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_le_eq
tff(fact_1541_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_1542_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_1543_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_1544_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_1545_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_1546_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_1547_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_1548_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_1549_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_1550_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).

% of_nat_0_le_iff
tff(fact_1551_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_1552_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_1553_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_1554_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_1555_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_1556_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_1557_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_1558_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_1559_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1560_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_1561_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_1562_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1563_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_1564_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1565_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_1566_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_1567_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1568_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1569_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1570_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1571_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1572_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_1573_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_1574_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_1575_eq__nat__nat__iff,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z3))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z3) )
        <=> ( Z = Z3 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_1576_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X5))) ) ) ).

% all_nat
tff(fact_1577_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X5))) ) ) ).

% ex_nat
tff(fact_1578_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_1579_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_1580_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_1581_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z))
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_1582_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_1583_le__of__int__ceiling,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ).

% le_of_int_ceiling
tff(fact_1584_le__of__int__ceiling,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ).

% le_of_int_ceiling
tff(fact_1585_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))))) ).

% real_nat_ceiling_ge
tff(fact_1586_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1587_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1588_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1589_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1590_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1591_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1592_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1593_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1594_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1595_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1596_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1597_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1598_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_1599_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_1600_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_1601_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_1602_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1603_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1604_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1605_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1606_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_1607_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_1608_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_1609_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_1610_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_1611_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_1612_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_1613_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_1614_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_1615_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_1616_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_1617_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_1618_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_1619_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_1620_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% le_imp_inverse_le_neg
tff(fact_1621_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% le_imp_inverse_le_neg
tff(fact_1622_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% inverse_le_imp_le_neg
tff(fact_1623_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% inverse_le_imp_le_neg
tff(fact_1624_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% le_imp_inverse_le
tff(fact_1625_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% le_imp_inverse_le
tff(fact_1626_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% inverse_le_imp_le
tff(fact_1627_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% inverse_le_imp_le
tff(fact_1628_inverse__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,X)),one_one_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% inverse_le_1_iff
tff(fact_1629_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% inverse_le_1_iff
tff(fact_1630_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_1631_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_1632_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_1633_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_1634_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ) ).

% of_nat_diff
tff(fact_1635_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_1636_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_1637_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_1638_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_1639_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_1640_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_1641_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% add1_zle_eq
tff(fact_1642_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1643_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_1644_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_1645_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))) ).

% ceiling_add_le
tff(fact_1646_ceiling__add__le,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).

% ceiling_add_le
tff(fact_1647_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
           => ( pp(aa_int_bool(P,I2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_1648_of__nat__ceiling,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2))))) ).

% of_nat_ceiling
tff(fact_1649_of__nat__ceiling,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R2))))) ).

% of_nat_ceiling
tff(fact_1650_dbl__inc__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_inc_def
tff(fact_1651_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_1652_dbl__inc__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_1653_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_1654_one__le__inverse__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,X)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% one_le_inverse_iff
tff(fact_1655_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ) ).

% one_le_inverse_iff
tff(fact_1656_inverse__less__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,X)),one_one_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% inverse_less_1_iff
tff(fact_1657_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% inverse_less_1_iff
tff(fact_1658_one__le__inverse,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A))) ) ) ).

% one_le_inverse
tff(fact_1659_one__le__inverse,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% one_le_inverse
tff(fact_1660_floor__exists1,axiom,
    ! [X: real] :
    ? [X3: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_1661_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_1662_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_1663_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_1664_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real))) ).

% of_int_ceiling_le_add_one
tff(fact_1665_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R2),one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_1666_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_1667_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_1668_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_1669_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_1670_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_1671_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_1672_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_1673_nat__add__distrib,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z3))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) ) ) ) ).

% nat_add_distrib
tff(fact_1674_nat__diff__distrib,axiom,
    ! [Z3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z3),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) ) ) ) ).

% nat_diff_distrib
tff(fact_1675_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_1676_ceiling__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I3: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,I3)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I3))) )
         => pp(aa_int_bool(P,I3)) ) ) ).

% ceiling_split
tff(fact_1677_ceiling__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim2889992004027027881ng_rat,T)))
    <=> ! [I3: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I3))) )
         => pp(aa_int_bool(P,I3)) ) ) ).

% ceiling_split
tff(fact_1678_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))) ) ) ).

% ceiling_eq_iff
tff(fact_1679_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))) ) ) ).

% ceiling_eq_iff
tff(fact_1680_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_1681_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_1682_ceiling__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).

% ceiling_correct
tff(fact_1683_ceiling__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ) ).

% ceiling_correct
tff(fact_1684_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% le_ceiling_iff
tff(fact_1685_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).

% le_ceiling_iff
tff(fact_1686_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% ceiling_less_iff
tff(fact_1687_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_1688_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1689_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_1690_dbl__dec__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_dec_def
tff(fact_1691_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_1692_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_1693_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_1694_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_1695_discrete,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).

% discrete
tff(fact_1696_discrete,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).

% discrete
tff(fact_1697_conj__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_1698_imp__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_1699_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1700_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1701_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1702_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1703_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1704_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1705_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1706_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1707_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1708_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1709_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1710_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1711_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1712_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1713_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1714_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B)),C))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1715_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B)),C))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1716_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1717_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1718_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1719_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1720_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1721_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_1722_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_1723_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_1724_abs__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_abs
tff(fact_1725_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_1726_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_1727_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_1728_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_1729_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_1730_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_1731_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_1732_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_1733_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_1734_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_1735_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_1736_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_1737_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_1738_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_1739_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_1740_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_1741_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_1742_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_1743_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_1744_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_1745_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_1746_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_1747_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_1748_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_1749_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_1750_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_1751_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_1752_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_1753_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_1754_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_1755_abs__minus,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
tff(fact_1756_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_1757_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_1758_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_1759_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_1760_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_1761_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_1762_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_1763_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_1764_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_1765_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_1766_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_1767_abs__inverse,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ).

% abs_inverse
tff(fact_1768_abs__inverse,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(abs_abs_complex,A)) ).

% abs_inverse
tff(fact_1769_abs__inverse,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ).

% abs_inverse
tff(fact_1770_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_1771_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_1772_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_1773_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_1774_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% abs_le_self_iff
tff(fact_1775_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_1776_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% abs_le_self_iff
tff(fact_1777_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_1778_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_1779_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_1780_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_1781_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_1782_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_1783_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_1784_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_1785_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_1786_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_1787_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_1788_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_1789_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_1790_of__int__abs,axiom,
    ! [X: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X)) ).

% of_int_abs
tff(fact_1791_of__int__abs,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,X)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,X)) ).

% of_int_abs
tff(fact_1792_of__int__abs,axiom,
    ! [X: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,X)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).

% of_int_abs
tff(fact_1793_of__int__abs,axiom,
    ! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,X)) ).

% of_int_abs
tff(fact_1794_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1795_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_1796_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1797_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1798_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_1799_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X4: real] : member_real(X4,S3)
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( member_real(X3,S3)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z4)) )
       => ? [Y3: real] :
            ( ! [X4: real] :
                ( member_real(X4,S3)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y3)) )
            & ! [Z4: real] :
                ( ! [X3: real] :
                    ( member_real(X3,S3)
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z4)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z4)) ) ) ) ) ).

% complete_real
tff(fact_1800_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => ? [X3: nat] :
            ( pp(aa_nat_bool(P,X3))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_1801_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_1802_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_1803_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_1804_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_1805_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_1806_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% abs_le_D1
tff(fact_1807_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% abs_le_D1
tff(fact_1808_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% abs_le_D1
tff(fact_1809_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% abs_le_D1
tff(fact_1810_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_1811_abs__ge__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A))) ).

% abs_ge_self
tff(fact_1812_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_self
tff(fact_1813_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_1814_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_1815_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_1816_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_1817_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_1818_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_1819_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_1820_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_1821_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_1822_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)) ).

% abs_minus_commute
tff(fact_1823_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_1824_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_1825_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_1826_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_1827_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_1828_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_1829_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,X) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_1830_abs__ge__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))) ).

% abs_ge_zero
tff(fact_1831_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_1832_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_zero
tff(fact_1833_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_1834_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_1835_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_1836_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_1837_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_1838_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_1839_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_1840_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_1841_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_1842_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq
tff(fact_1843_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1844_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1845_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1846_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))) ).

% abs_triangle_ineq2
tff(fact_1847_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1848_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_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(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq2
tff(fact_1849_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1850_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))),abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))) ).

% abs_triangle_ineq3
tff(fact_1851_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1852_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_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(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq3
tff(fact_1853_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1854_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_1855_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1856_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_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(minus_minus_rat(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_1857_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1858_abs__leI,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).

% abs_leI
tff(fact_1859_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_1860_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).

% abs_leI
tff(fact_1861_abs__leI,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).

% abs_leI
tff(fact_1862_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).

% abs_le_D2
tff(fact_1863_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% abs_le_D2
tff(fact_1864_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).

% abs_le_D2
tff(fact_1865_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% abs_le_D2
tff(fact_1866_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_le_iff
tff(fact_1867_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_le_iff
tff(fact_1868_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_le_iff
tff(fact_1869_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_le_iff
tff(fact_1870_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_1871_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_1872_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_minus_self
tff(fact_1873_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_1874_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_less_iff
tff(fact_1875_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_less_iff
tff(fact_1876_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_less_iff
tff(fact_1877_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_less_iff
tff(fact_1878_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_1879_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_1880_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_1881_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_1882_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1883_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_1884_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1885_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1886_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1887_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1888_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1889_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1890_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1891_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1892_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1893_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1894_abs__if__raw,axiom,
    ! [X4: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X4) = aa_int_int(uminus_uminus_int,X4) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_1895_abs__if__raw,axiom,
    ! [X4: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X4) = aa_real_real(uminus_uminus_real,X4) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_1896_abs__if__raw,axiom,
    ! [X4: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X4) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_1897_abs__if__raw,axiom,
    ! [X4: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X4) = aa_rat_rat(uminus_uminus_rat,X4) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X4) = X4 ) ) ) ).

% abs_if_raw
tff(fact_1898_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_1899_abs__of__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1900_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_1901_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_1902_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_1903_abs__if,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_if
tff(fact_1904_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = A ) ) ) ).

% abs_if
tff(fact_1905_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_if
tff(fact_1906_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq4
tff(fact_1907_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1908_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_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_1909_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1910_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C))),abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_1911_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_1912_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_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(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_1913_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_1914_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_1915_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_1916_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_1917_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_1918_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_1919_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_1920_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_1921_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_1922_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_1923_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1924_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).

% minf(7)
tff(fact_1925_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).

% minf(7)
tff(fact_1926_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X4)) ) ).

% minf(7)
tff(fact_1927_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).

% minf(7)
tff(fact_1928_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).

% minf(7)
tff(fact_1929_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).

% minf(5)
tff(fact_1930_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).

% minf(5)
tff(fact_1931_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),T)) ) ).

% minf(5)
tff(fact_1932_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).

% minf(5)
tff(fact_1933_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).

% minf(5)
tff(fact_1934_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1935_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1936_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1937_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1938_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_1939_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1940_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1941_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1942_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1943_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_1944_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z4))
         => ( pp(aa_real_bool(P,X3))
          <=> pp(aa_real_bool(P2,X3)) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z4))
           => ( pp(aa_real_bool(Q,X3))
            <=> pp(aa_real_bool(Q3,X3)) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
           => ( ( pp(aa_real_bool(P,X4))
                | pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                | pp(aa_real_bool(Q3,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1945_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z4))
         => ( pp(aa_rat_bool(P,X3))
          <=> pp(aa_rat_bool(P2,X3)) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z4))
           => ( pp(aa_rat_bool(Q,X3))
            <=> pp(aa_rat_bool(Q3,X3)) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
           => ( ( pp(aa_rat_bool(P,X4))
                | pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                | pp(aa_rat_bool(Q3,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1946_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z4))
         => ( pp(aa_num_bool(P,X3))
          <=> pp(aa_num_bool(P2,X3)) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z4))
           => ( pp(aa_num_bool(Q,X3))
            <=> pp(aa_num_bool(Q3,X3)) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
           => ( ( pp(aa_num_bool(P,X4))
                | pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                | pp(aa_num_bool(Q3,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1947_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z4))
         => ( pp(aa_nat_bool(P,X3))
          <=> pp(aa_nat_bool(P2,X3)) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z4))
           => ( pp(aa_nat_bool(Q,X3))
            <=> pp(aa_nat_bool(Q3,X3)) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
           => ( ( pp(aa_nat_bool(P,X4))
                | pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                | pp(aa_nat_bool(Q3,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1948_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
         => ( pp(aa_int_bool(P,X3))
          <=> pp(aa_int_bool(P2,X3)) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
           => ( pp(aa_int_bool(Q,X3))
            <=> pp(aa_int_bool(Q3,X3)) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                | pp(aa_int_bool(Q3,X4)) ) ) ) ) ) ).

% minf(2)
tff(fact_1949_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z4))
         => ( pp(aa_real_bool(P,X3))
          <=> pp(aa_real_bool(P2,X3)) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z4))
           => ( pp(aa_real_bool(Q,X3))
            <=> pp(aa_real_bool(Q3,X3)) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
           => ( ( pp(aa_real_bool(P,X4))
                & pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                & pp(aa_real_bool(Q3,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1950_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z4))
         => ( pp(aa_rat_bool(P,X3))
          <=> pp(aa_rat_bool(P2,X3)) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z4))
           => ( pp(aa_rat_bool(Q,X3))
            <=> pp(aa_rat_bool(Q3,X3)) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
           => ( ( pp(aa_rat_bool(P,X4))
                & pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                & pp(aa_rat_bool(Q3,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1951_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z4))
         => ( pp(aa_num_bool(P,X3))
          <=> pp(aa_num_bool(P2,X3)) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z4))
           => ( pp(aa_num_bool(Q,X3))
            <=> pp(aa_num_bool(Q3,X3)) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
           => ( ( pp(aa_num_bool(P,X4))
                & pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                & pp(aa_num_bool(Q3,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1952_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z4))
         => ( pp(aa_nat_bool(P,X3))
          <=> pp(aa_nat_bool(P2,X3)) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z4))
           => ( pp(aa_nat_bool(Q,X3))
            <=> pp(aa_nat_bool(Q3,X3)) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
           => ( ( pp(aa_nat_bool(P,X4))
                & pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                & pp(aa_nat_bool(Q3,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1953_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
         => ( pp(aa_int_bool(P,X3))
          <=> pp(aa_int_bool(P2,X3)) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
           => ( pp(aa_int_bool(Q,X3))
            <=> pp(aa_int_bool(Q3,X3)) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                & pp(aa_int_bool(Q3,X4)) ) ) ) ) ) ).

% minf(1)
tff(fact_1954_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).

% pinf(7)
tff(fact_1955_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).

% pinf(7)
tff(fact_1956_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X4)) ) ).

% pinf(7)
tff(fact_1957_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).

% pinf(7)
tff(fact_1958_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).

% pinf(7)
tff(fact_1959_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).

% pinf(5)
tff(fact_1960_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).

% pinf(5)
tff(fact_1961_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),T)) ) ).

% pinf(5)
tff(fact_1962_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).

% pinf(5)
tff(fact_1963_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).

% pinf(5)
tff(fact_1964_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1965_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1966_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1967_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1968_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_1969_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1970_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1971_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1972_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1973_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_1974_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X3))
         => ( pp(aa_real_bool(P,X3))
          <=> pp(aa_real_bool(P2,X3)) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X3))
           => ( pp(aa_real_bool(Q,X3))
            <=> pp(aa_real_bool(Q3,X3)) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
           => ( ( pp(aa_real_bool(P,X4))
                | pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                | pp(aa_real_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1975_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X3))
         => ( pp(aa_rat_bool(P,X3))
          <=> pp(aa_rat_bool(P2,X3)) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X3))
           => ( pp(aa_rat_bool(Q,X3))
            <=> pp(aa_rat_bool(Q3,X3)) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
           => ( ( pp(aa_rat_bool(P,X4))
                | pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                | pp(aa_rat_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1976_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X3))
         => ( pp(aa_num_bool(P,X3))
          <=> pp(aa_num_bool(P2,X3)) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X3))
           => ( pp(aa_num_bool(Q,X3))
            <=> pp(aa_num_bool(Q3,X3)) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
           => ( ( pp(aa_num_bool(P,X4))
                | pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                | pp(aa_num_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1977_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X3))
         => ( pp(aa_nat_bool(P,X3))
          <=> pp(aa_nat_bool(P2,X3)) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X3))
           => ( pp(aa_nat_bool(Q,X3))
            <=> pp(aa_nat_bool(Q3,X3)) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
           => ( ( pp(aa_nat_bool(P,X4))
                | pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                | pp(aa_nat_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1978_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
         => ( pp(aa_int_bool(P,X3))
          <=> pp(aa_int_bool(P2,X3)) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
           => ( pp(aa_int_bool(Q,X3))
            <=> pp(aa_int_bool(Q3,X3)) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                | pp(aa_int_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1979_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X3))
         => ( pp(aa_real_bool(P,X3))
          <=> pp(aa_real_bool(P2,X3)) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X3))
           => ( pp(aa_real_bool(Q,X3))
            <=> pp(aa_real_bool(Q3,X3)) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
           => ( ( pp(aa_real_bool(P,X4))
                & pp(aa_real_bool(Q,X4)) )
            <=> ( pp(aa_real_bool(P2,X4))
                & pp(aa_real_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1980_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X3))
         => ( pp(aa_rat_bool(P,X3))
          <=> pp(aa_rat_bool(P2,X3)) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X3))
           => ( pp(aa_rat_bool(Q,X3))
            <=> pp(aa_rat_bool(Q3,X3)) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
           => ( ( pp(aa_rat_bool(P,X4))
                & pp(aa_rat_bool(Q,X4)) )
            <=> ( pp(aa_rat_bool(P2,X4))
                & pp(aa_rat_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1981_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X3))
         => ( pp(aa_num_bool(P,X3))
          <=> pp(aa_num_bool(P2,X3)) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X3))
           => ( pp(aa_num_bool(Q,X3))
            <=> pp(aa_num_bool(Q3,X3)) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
           => ( ( pp(aa_num_bool(P,X4))
                & pp(aa_num_bool(Q,X4)) )
            <=> ( pp(aa_num_bool(P2,X4))
                & pp(aa_num_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1982_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X3))
         => ( pp(aa_nat_bool(P,X3))
          <=> pp(aa_nat_bool(P2,X3)) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X3))
           => ( pp(aa_nat_bool(Q,X3))
            <=> pp(aa_nat_bool(Q3,X3)) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
           => ( ( pp(aa_nat_bool(P,X4))
                & pp(aa_nat_bool(Q,X4)) )
            <=> ( pp(aa_nat_bool(P2,X4))
                & pp(aa_nat_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1983_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
         => ( pp(aa_int_bool(P,X3))
          <=> pp(aa_int_bool(P2,X3)) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
           => ( pp(aa_int_bool(Q,X3))
            <=> pp(aa_int_bool(Q3,X3)) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
            <=> ( pp(aa_int_bool(P2,X4))
                & pp(aa_int_bool(Q3,X4)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1984_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_1985_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X)) ).

% lt_ex
tff(fact_1986_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X)) ).

% lt_ex
tff(fact_1987_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_12)) ).

% gt_ex
tff(fact_1988_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_12)) ).

% gt_ex
tff(fact_1989_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_12)) ).

% gt_ex
tff(fact_1990_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_12)) ).

% gt_ex
tff(fact_1991_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y)) ) ) ).

% dense
tff(fact_1992_dense,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ? [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y)) ) ) ).

% dense
tff(fact_1993_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1994_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1995_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1996_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1997_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_1998_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order.asym
tff(fact_1999_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order.asym
tff(fact_2000_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order.asym
tff(fact_2001_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order.asym
tff(fact_2002_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order.asym
tff(fact_2003_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_2004_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_2005_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_2006_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_2007_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_2008_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_2009_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_2010_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_2011_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_2012_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_2013_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X3))
             => pp(aa_nat_bool(P,Y4)) )
         => pp(aa_nat_bool(P,X3)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_2014_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2015_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2016_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2017_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2018_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_2019_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_2020_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ( X != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_cases
tff(fact_2021_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_2022_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_2023_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_2024_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% dual_order.asym
tff(fact_2025_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% dual_order.asym
tff(fact_2026_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% dual_order.asym
tff(fact_2027_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% dual_order.asym
tff(fact_2028_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% dual_order.asym
tff(fact_2029_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_2030_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% dual_order.irrefl
tff(fact_2031_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% dual_order.irrefl
tff(fact_2032_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_2033_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_2034_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [N4: nat] :
          ( pp(aa_nat_bool(P,N4))
          & ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N4))
             => ~ pp(aa_nat_bool(P,M4)) ) ) ) ).

% exists_least_iff
tff(fact_2035_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_2036_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),A3))
       => ( ! [A3: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_2037_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_2038_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_2039_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_2040_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_2041_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans
tff(fact_2042_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans
tff(fact_2043_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_2044_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_2045_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2046_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2047_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2048_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2049_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_2050_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_2051_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_2052_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_2053_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_2054_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_2055_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_2056_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_2057_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_2058_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_2059_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_2060_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_2061_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_2062_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_2063_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_2064_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_2065_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE
tff(fact_2066_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_2067_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_2068_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_2069_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_2070_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_asym
tff(fact_2071_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_asym
tff(fact_2072_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_2073_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_2074_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_2075_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_2076_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_2077_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_2078_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_2079_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_2080_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order_less_asym'
tff(fact_2081_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order_less_asym'
tff(fact_2082_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order_less_asym'
tff(fact_2083_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order_less_asym'
tff(fact_2084_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order_less_asym'
tff(fact_2085_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_trans
tff(fact_2086_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_trans
tff(fact_2087_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_trans
tff(fact_2088_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_trans
tff(fact_2089_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_trans
tff(fact_2090_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2091_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2092_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2093_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2094_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2095_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2096_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2097_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2098_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2099_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_2100_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2101_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2102_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2103_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2104_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2105_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2106_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2107_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2108_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2109_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_2110_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_2111_order__less__irrefl,axiom,
    ! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).

% order_less_irrefl
tff(fact_2112_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_2113_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_2114_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_2115_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2116_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2117_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2118_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2119_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2120_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2121_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2122_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2123_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2124_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_2125_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2126_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2127_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2128_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2129_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B)),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2130_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2131_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2132_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2133_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2134_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_2135_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_not_sym
tff(fact_2136_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_not_sym
tff(fact_2137_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_2138_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_2139_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_2140_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_2141_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_2142_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_2143_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_2144_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_2145_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_2146_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      | ( X = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_less_linear
tff(fact_2147_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_2148_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_2149_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_2150_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2151_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2152_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2153_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2154_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_2155_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2156_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2157_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2158_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2159_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2160_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_2161_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_2162_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_2163_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_2164_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_2165_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B2)
    <=> ! [X5: complex] :
          ( member_complex(X5,set_complex2(Xs))
         => member_complex(X5,B2) ) ) ).

% subset_code(1)
tff(fact_2166_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B2)
    <=> ! [X5: real] :
          ( member_real(X5,set_real2(Xs))
         => member_real(X5,B2) ) ) ).

% subset_code(1)
tff(fact_2167_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B2)
    <=> ! [X5: set_nat] :
          ( member_set_nat(X5,set_set_nat2(Xs))
         => member_set_nat(X5,B2) ) ) ).

% subset_code(1)
tff(fact_2168_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),B2)) ) ) ).

% subset_code(1)
tff(fact_2169_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B2)
    <=> ! [X5: int] :
          ( member_int(X5,set_int2(Xs))
         => member_int(X5,B2) ) ) ).

% subset_code(1)
tff(fact_2170_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),B2))
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X5),B2)) ) ) ).

% subset_code(1)
tff(fact_2171_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),abs_abs_Code_integer(X)))) ).

% abs_add_one_gt_zero
tff(fact_2172_abs__add__one__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_2173_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).

% abs_add_one_gt_zero
tff(fact_2174_abs__add__one__gt__zero,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_2175_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% of_int_leD
tff(fact_2176_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).

% of_int_leD
tff(fact_2177_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% of_int_leD
tff(fact_2178_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).

% of_int_leD
tff(fact_2179_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).

% of_int_lessD
tff(fact_2180_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% of_int_lessD
tff(fact_2181_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% of_int_lessD
tff(fact_2182_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).

% of_int_lessD
tff(fact_2183_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_2184_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2185_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2186_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).

% minf(8)
tff(fact_2187_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).

% minf(8)
tff(fact_2188_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X4)) ) ).

% minf(8)
tff(fact_2189_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).

% minf(8)
tff(fact_2190_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).

% minf(8)
tff(fact_2191_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).

% minf(6)
tff(fact_2192_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).

% minf(6)
tff(fact_2193_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),T)) ) ).

% minf(6)
tff(fact_2194_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).

% minf(6)
tff(fact_2195_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).

% minf(6)
tff(fact_2196_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).

% pinf(8)
tff(fact_2197_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).

% pinf(8)
tff(fact_2198_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X4)) ) ).

% pinf(8)
tff(fact_2199_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).

% pinf(8)
tff(fact_2200_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).

% pinf(8)
tff(fact_2201_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).

% pinf(6)
tff(fact_2202_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).

% pinf(6)
tff(fact_2203_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X4))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),T)) ) ).

% pinf(6)
tff(fact_2204_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).

% pinf(6)
tff(fact_2205_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).

% pinf(6)
tff(fact_2206_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_2207_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y)) ) ).

% leD
tff(fact_2208_leD,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% leD
tff(fact_2209_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_2210_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_2211_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_2212_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% leI
tff(fact_2213_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% leI
tff(fact_2214_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_2215_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_2216_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_2217_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2218_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2219_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2220_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2221_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2222_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2223_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2224_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2225_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2226_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2227_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2228_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2229_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2230_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2231_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2232_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2233_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2234_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2235_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_ge
tff(fact_2236_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_ge
tff(fact_2237_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_le
tff(fact_2238_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_le
tff(fact_2239_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2240_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).

% less_le_not_le
tff(fact_2241_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2242_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2243_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2244_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2245_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_2246_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% not_le_imp_less
tff(fact_2247_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_2248_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_2249_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_2250_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2251_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2252_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2253_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2254_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2255_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2256_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2257_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2258_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2259_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2260_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2261_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2262_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_2263_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).

% order.strict_trans1
tff(fact_2264_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_2265_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans1
tff(fact_2266_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_2267_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_2268_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_2269_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).

% order.strict_trans2
tff(fact_2270_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_2271_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans2
tff(fact_2272_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_2273_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_2274_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_2275_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).

% order.strict_iff_not
tff(fact_2276_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_2277_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_2278_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_2279_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_2280_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
     => ( ! [W2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W2)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_2281_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
     => ( ! [W2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),X))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W2)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_2282_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_2283_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ! [W2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Y))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_2284_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2285_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2286_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2287_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2288_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2289_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2290_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2291_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2292_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2293_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2294_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2295_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2296_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_2297_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_2298_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_2299_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_2300_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_2301_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_2302_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_2303_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_2304_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_2305_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_2306_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_2307_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_2308_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_2309_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_2310_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_2311_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_2312_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_2313_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_2314_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% order.strict_implies_order
tff(fact_2315_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% order.strict_implies_order
tff(fact_2316_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% order.strict_implies_order
tff(fact_2317_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% order.strict_implies_order
tff(fact_2318_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% order.strict_implies_order
tff(fact_2319_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% order.strict_implies_order
tff(fact_2320_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_2321_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_2322_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_2323_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_2324_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_2325_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_2326_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2327_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2328_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2329_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2330_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2331_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2332_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2333_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2334_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2335_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2336_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2337_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2338_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_not_le
tff(fact_2339_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_not_le
tff(fact_2340_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_2341_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_2342_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_2343_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_2344_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_not_less
tff(fact_2345_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_2346_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_2347_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_2348_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_2349_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% order_less_imp_le
tff(fact_2350_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_less_imp_le
tff(fact_2351_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_2352_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_2353_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_2354_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ( A != B )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_2355_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( ( A != B )
       => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).

% order_le_neq_trans
tff(fact_2356_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( A != B )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_2357_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( A != B )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_2358_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( A != B )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_2359_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( A != B )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_2360_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_2361_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).

% order_neq_le_trans
tff(fact_2362_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_2363_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_2364_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_2365_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_2366_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_2367_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(Y),Z))
       => pp(aa_set_nat_bool(ord_less_set_nat(X),Z)) ) ) ).

% order_le_less_trans
tff(fact_2368_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_2369_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_2370_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_2371_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_2372_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_2373_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z))
       => pp(aa_set_nat_bool(ord_less_set_nat(X),Z)) ) ) ).

% order_less_le_trans
tff(fact_2374_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_2375_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_2376_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_2377_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_2378_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2379_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2380_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2381_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2382_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2383_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2384_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2385_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2386_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2387_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_2388_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2389_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2390_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2391_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2392_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2393_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2394_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2395_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2396_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2397_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_2398_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2399_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2400_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2401_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2402_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2403_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2404_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2405_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2406_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2407_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_2408_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
         => pp(aa_nat_bool(P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
           => ( ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I4))
                 => pp(aa_nat_bool(P,I4)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_2409_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B))
         => ? [C2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
              & ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),C2)) )
                 => pp(aa_real_bool(P,X4)) )
              & ! [D5: real] :
                  ( ! [X3: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),D5)) )
                     => pp(aa_real_bool(P,X3)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D5),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_2410_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B))
         => ? [C2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
              & ! [X4: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),C2)) )
                 => pp(aa_nat_bool(P,X4)) )
              & ! [D5: nat] :
                  ( ! [X3: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),D5)) )
                     => pp(aa_nat_bool(P,X3)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D5),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_2411_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B))
         => ? [C2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
              & ! [X4: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),C2)) )
                 => pp(aa_int_bool(P,X4)) )
              & ! [D5: int] :
                  ( ! [X3: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),D5)) )
                     => pp(aa_int_bool(P,X3)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D5),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_2412_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_2413_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_2414_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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_2415_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X22: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X22)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_2416_bset_I6_J,axiom,
    ! [D6: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,B2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6)),T)) ) ) ) ).

% bset(6)
tff(fact_2417_bset_I8_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6))) ) ) ) ) ).

% bset(8)
tff(fact_2418_aset_I6_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6)),T)) ) ) ) ) ).

% aset(6)
tff(fact_2419_aset_I8_J,axiom,
    ! [D6: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6))) ) ) ) ).

% aset(8)
tff(fact_2420_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2421_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2422_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_2423_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_2424_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2425_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2426_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_2427_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_2428_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2429_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2430_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_2431_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_2432_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2433_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2434_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_2435_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_2436_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_2437_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_2438_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_2439_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_2440_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2441_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_2442_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2443_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_2444_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2445_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_2446_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_2447_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_2448_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_2449_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_2450_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2451_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_2452_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2453_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2454_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_2455_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2456_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2457_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_2458_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2459_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2460_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_2461_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2462_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2463_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_2464_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_2465_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_2466_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_2467_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2468_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2469_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2470_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_2471_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_2472_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_2473_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_2474_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_2475_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_2476_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_2477_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_2478_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_2479_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_2480_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_2481_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_2482_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_2483_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_2484_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_2485_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_2486_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_2487_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_2488_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_2489_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_2490_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_2491_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_2492_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_2493_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_2494_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_2495_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mult
tff(fact_2496_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_2497_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_2498_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_2499_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_2500_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_2501_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_2502_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_2503_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).

% inverse_mult_distrib
tff(fact_2504_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(B),A) ).

% inverse_divide
tff(fact_2505_inverse__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(B),A) ).

% inverse_divide
tff(fact_2506_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(B),A) ).

% inverse_divide
tff(fact_2507_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_2508_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2509_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2510_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_2511_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_2512_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2513_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2514_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_2515_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_2516_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2517_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2518_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_2519_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_2520_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2521_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2522_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_2523_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2524_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2525_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2526_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_2527_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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_2528_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2529_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2530_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2531_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_2532_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_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_2533_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2534_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2535_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2536_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2537_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2538_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2539_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2540_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2541_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2542_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2543_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2544_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2545_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2546_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2547_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2548_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2549_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_2550_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2551_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_2552_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2553_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2554_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_2555_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2556_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2557_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_2558_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2559_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2560_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2561_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2562_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_2563_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(divide_divide_real(A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_2564_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_2565_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_2566_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2567_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_2568_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2569_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_2570_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2571_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_2572_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2573_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_2574_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_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_2575_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_2576_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_2577_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_2578_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_2579_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_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_2580_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_2581_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_2582_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_2583_divide__minus1,axiom,
    ! [X: real] : aa_real_real(divide_divide_real(X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_2584_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_2585_divide__minus1,axiom,
    ! [X: rat] : aa_rat_rat(divide_divide_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).

% divide_minus1
tff(fact_2586_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_mult
tff(fact_2587_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_mult
tff(fact_2588_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_2589_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_2590_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_2591_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_2592_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_2593_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2594_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2595_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2596_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2597_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2598_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2599_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2600_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2601_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2602_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2603_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2604_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2605_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_2606_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_2607_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2608_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2609_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2610_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2611_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2612_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2613_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2614_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2615_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2616_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2617_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_2618_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_2619_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_2620_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_2621_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_2622_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_2623_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2624_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2625_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2626_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2627_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2628_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2629_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2630_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2631_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_2632_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% field_class.field_divide_inverse
tff(fact_2633_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% field_class.field_divide_inverse
tff(fact_2634_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_2635_divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% divide_inverse
tff(fact_2636_divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% divide_inverse
tff(fact_2637_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_2638_divide__inverse__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).

% divide_inverse_commute
tff(fact_2639_divide__inverse__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),A) ).

% divide_inverse_commute
tff(fact_2640_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.left_commute
tff(fact_2641_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_2642_mult_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)) ).

% mult.left_commute
tff(fact_2643_mult_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)) ).

% mult.left_commute
tff(fact_2644_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% mult.commute
tff(fact_2645_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_2646_mult_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) ).

% mult.commute
tff(fact_2647_mult_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) ).

% mult.commute
tff(fact_2648_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_2649_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_2650_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_2651_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_2652_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2653_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2654_ab__semigroup__mult__class_Omult__ac_I1_J,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)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2655_ab__semigroup__mult__class_Omult__ac_I1_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(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)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2656_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2657_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(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2658_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2659_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2660_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(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2661_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2662_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2663_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(divide_divide_real(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2664_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2665_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2666_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(divide_divide_real(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2667_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2668_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_2669_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(divide_divide_real(B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_2670_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_2671_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_2672_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),C) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_2673_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_2674_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2675_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(X),Y) = aa_real_real(divide_divide_real(W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2676_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2677_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2678_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2679_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2680_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2681_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2682_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2683_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2684_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2685_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2686_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2687_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2688_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2689_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2690_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2691_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2692_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2693_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2694_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2695_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2696_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2697_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2698_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2699_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2700_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(X),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2701_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2702_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2703_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2704_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2705_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2706_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(divide_divide_real(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2707_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2708_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2709_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(X),Y)),Z) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2710_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2711_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2712_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: 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(X),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(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2713_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2714_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2715_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(divide_divide_real(B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(divide_divide_real(B),Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2716_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2717_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2718_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(A),Z)),B) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2719_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2720_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2721_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2722_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2723_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2724_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2725_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2726_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2727_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2728_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2729_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2730_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2731_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2732_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)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_2733_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_2734_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_2735_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 )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_2736_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_2737_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_2738_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(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2739_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2740_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(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2741_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(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2742_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2743_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(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2744_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_2745_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_2746_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_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(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_2747_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_2748_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_2749_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_2750_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_2751_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_2752_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_2753_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_2754_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_2755_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_2756_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_2757_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_2758_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_2759_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2760_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2761_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_2762_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_2763_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2764_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2765_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_2766_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_2767_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2768_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2769_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_2770_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_2771_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2772_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2773_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_2774_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_2775_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2776_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2777_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_2778_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_2779_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2780_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2781_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2782_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_2783_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_2784_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2785_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2786_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2787_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_2788_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_2789_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2790_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2791_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_2792_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2793_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_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(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2794_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_2795_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2796_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2797_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_2798_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_2799_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_2800_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_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(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_2801_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_2802_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_2803_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_2804_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_2805_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_2806_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_2807_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2808_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2809_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2810_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2811_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2812_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2813_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2814_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2815_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2816_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2817_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_2818_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_2819_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_2820_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_2821_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_2822_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_2823_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_2824_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_2825_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),C)),A) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_2826_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_2827_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_2828_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_2829_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_2830_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_2831_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_2832_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_2833_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D6: real,Q: fun_real_bool] :
      ( ! [X3: real,K2: real] :
          ( pp(aa_real_bool(P,X3))
        <=> pp(aa_real_bool(P,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D6)))) )
     => ( ! [X3: real,K2: real] :
            ( pp(aa_real_bool(Q,X3))
          <=> pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D6)))) )
       => ! [X4: real,K4: real] :
            ( ( pp(aa_real_bool(P,X4))
              | pp(aa_real_bool(Q,X4)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(minus_minus_real(X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D6))))
              | pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D6)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2834_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D6: rat,Q: fun_rat_bool] :
      ( ! [X3: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X3))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D6)))) )
     => ( ! [X3: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X3))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D6)))) )
       => ! [X4: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X4))
              | pp(aa_rat_bool(Q,X4)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K4),D6))))
              | pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K4),D6)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2835_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D6: int,Q: fun_int_bool] :
      ( ! [X3: int,K2: int] :
          ( pp(aa_int_bool(P,X3))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6)))) )
     => ( ! [X3: int,K2: int] :
            ( pp(aa_int_bool(Q,X3))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6)))) )
       => ! [X4: int,K4: int] :
            ( ( pp(aa_int_bool(P,X4))
              | pp(aa_int_bool(Q,X4)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D6))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D6)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2836_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D6: real,Q: fun_real_bool] :
      ( ! [X3: real,K2: real] :
          ( pp(aa_real_bool(P,X3))
        <=> pp(aa_real_bool(P,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D6)))) )
     => ( ! [X3: real,K2: real] :
            ( pp(aa_real_bool(Q,X3))
          <=> pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D6)))) )
       => ! [X4: real,K4: real] :
            ( ( pp(aa_real_bool(P,X4))
              & pp(aa_real_bool(Q,X4)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(minus_minus_real(X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D6))))
              & pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D6)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2837_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D6: rat,Q: fun_rat_bool] :
      ( ! [X3: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X3))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D6)))) )
     => ( ! [X3: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X3))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D6)))) )
       => ! [X4: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X4))
              & pp(aa_rat_bool(Q,X4)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K4),D6))))
              & pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K4),D6)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2838_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D6: int,Q: fun_int_bool] :
      ( ! [X3: int,K2: int] :
          ( pp(aa_int_bool(P,X3))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6)))) )
     => ( ! [X3: int,K2: int] :
            ( pp(aa_int_bool(Q,X3))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6)))) )
       => ! [X4: int,K4: int] :
            ( ( pp(aa_int_bool(P,X4))
              & pp(aa_int_bool(Q,X4)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D6))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D6)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2839_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_2840_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_2841_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_2842_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_2843_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_2844_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_2845_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_2846_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 = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_2847_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_2848_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_2849_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_2850_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_2851_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_2852_mult__of__nat__commute,axiom,
    ! [X: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,X)) ).

% mult_of_nat_commute
tff(fact_2853_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_2854_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_2855_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_2856_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_2857_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_2858_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_2859_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_2860_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_2861_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,X: complex] :
      ( ( aa_complex_complex(times_times_complex(Y),X) = aa_complex_complex(times_times_complex(X),Y) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_2862_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( aa_rat_rat(times_times_rat(Y),X) = aa_rat_rat(times_times_rat(X),Y) )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Y)),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_2863_mult__of__int__commute,axiom,
    ! [X: int,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).

% mult_of_int_commute
tff(fact_2864_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,X)) ).

% mult_of_int_commute
tff(fact_2865_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_2866_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_2867_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_2868_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_2869_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_2870_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_2871_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_2872_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2873_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2874_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2875_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2876_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2877_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono
tff(fact_2878_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_2879_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_2880_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2881_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2882_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2883_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2884_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_2885_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_2886_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2887_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2888_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2889_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2890_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2891_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2892_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_2893_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_2894_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_2895_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_2896_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2897_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2898_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2899_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2900_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2901_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2902_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2903_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2904_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_2905_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_2906_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_2907_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_2908_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: 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(A),Z))),B) = B ) )
      & ( ( 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(A),Z))),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(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2909_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2910_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: 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(divide_divide_rat(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_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(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2911_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: 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(X),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2912_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2913_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2914_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: 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(X),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2915_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2916_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2917_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z)),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z)),B) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2918_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2919_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2920_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2921_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2922_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2923_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(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_2924_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_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(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_2925_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))
    <=> ( ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
          | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
            & ( pp(aa_set_nat_bool(ord_less_set_nat(C),A))
              | pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2926_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2927_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2928_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2929_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2930_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2931_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_2932_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_2933_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2934_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2935_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2936_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2937_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2938_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2939_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2940_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2941_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2942_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2943_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(minus_minus_real(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_2944_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_2945_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int,K2: int] :
            ( pp(aa_int_bool(P,X3))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
        <=> ? [X5: int] :
              ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D))
              & pp(aa_int_bool(P,X5)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2946_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le
tff(fact_2947_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_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_2948_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2949_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2950_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2951_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_right_mono
tff(fact_2952_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2953_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2954_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2955_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2956_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2957_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2958_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2959_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2960_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2961_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2962_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2963_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_2964_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(divide_divide_int(N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,I3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,I3)) ) ) ) ) ).

% split_zdiv
tff(fact_2965_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2966_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2967_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2968_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2969_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2970_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_2971_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2972_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2973_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2974_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2975_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_2976_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_2977_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2978_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2979_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_2980_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2981_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_2982_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2983_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2984_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2985_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2986_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_2987_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2988_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_2989_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_2990_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_2991_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_2992_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2993_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2994_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2995_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2996_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2997_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_2998_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2999_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_3000_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_3001_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3002_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3003_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3004_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3005_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_3006_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_3007_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3008_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3009_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_3010_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_3011_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3012_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3013_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_3014_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_3015_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3016_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3017_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_3018_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_3019_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3020_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_3021_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_3022_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_3023_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_3024_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono
tff(fact_3025_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono
tff(fact_3026_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3027_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3028_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_3029_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_3030_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3031_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono
tff(fact_3032_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono
tff(fact_3033_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3034_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3035_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_3036_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_3037_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3038_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_3039_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_3040_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3041_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ).

% split_mult_pos_le
tff(fact_3042_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ).

% split_mult_pos_le
tff(fact_3043_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_3044_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).

% zero_le_square
tff(fact_3045_zero__le__square,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A))) ).

% zero_le_square
tff(fact_3046_zero__le__square,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_3047_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_3048_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_3049_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3050_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3051_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_3052_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_3053_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3054_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3055_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3056_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3057_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3058_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3059_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3060_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3061_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3062_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_3063_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_3064_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3065_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3066_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3067_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3068_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3069_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3070_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3071_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3072_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_3073_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_3074_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3075_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3076_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3077_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3078_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3079_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3080_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3081_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3082_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3083_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3084_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3085_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_3086_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_3087_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_3088_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_3089_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos
tff(fact_3090_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_3091_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_3092_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos
tff(fact_3093_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_3094_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_3095_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_3096_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_3097_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_3098_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_3099_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_3100_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_pos_pos
tff(fact_3101_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_pos_pos
tff(fact_3102_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_3103_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_pos_pos
tff(fact_3104_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_3105_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_3106_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_3107_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_3108_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_3109_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_3110_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_3111_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_3112_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_3113_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_3114_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_3115_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_3116_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_3117_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_3118_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_neg_neg
tff(fact_3119_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_neg_neg
tff(fact_3120_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_neg_neg
tff(fact_3121_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_3122_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_3123_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3124_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3125_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != 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,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3126_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != 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,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3127_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N))) ) ) ).

% less_1_mult
tff(fact_3128_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N))) ) ) ).

% less_1_mult
tff(fact_3129_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% less_1_mult
tff(fact_3130_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).

% less_1_mult
tff(fact_3131_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3132_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3133_eq__add__iff1,axiom,
    ! [A: int,E2: 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),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),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)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3134_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3135_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3136_eq__add__iff2,axiom,
    ! [A: int,E2: 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),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),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)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3137_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(minus_minus_real(X),Y)) ).

% square_diff_square_factored
tff(fact_3138_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(minus_minus_rat(X),Y)) ).

% square_diff_square_factored
tff(fact_3139_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_3140_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(minus_minus_real(Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(X),A)),B)) ).

% mult_diff_mult
tff(fact_3141_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),A)),B)) ).

% mult_diff_mult
tff(fact_3142_mult__diff__mult,axiom,
    ! [X: 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,X),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,X),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,X),A)),B)) ).

% mult_diff_mult
tff(fact_3143_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_3144_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_3145_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_3146_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_3147_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_3148_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ).

% inverse_eq_divide
tff(fact_3149_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_3150_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ).

% inverse_eq_divide
tff(fact_3151_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D))) ) ) ).

% abs_mult_less
tff(fact_3152_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_3153_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D))) ) ) ).

% abs_mult_less
tff(fact_3154_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3155_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3156_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3157_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3158_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3159_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3160_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_3161_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_3162_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).

% inverse_unique
tff(fact_3163_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).

% inverse_unique
tff(fact_3164_mult__inverse__of__nat__commute,axiom,
    ! [Xa3: nat,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa3))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa3))) ).

% mult_inverse_of_nat_commute
tff(fact_3165_mult__inverse__of__nat__commute,axiom,
    ! [Xa3: nat,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa3))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa3))) ).

% mult_inverse_of_nat_commute
tff(fact_3166_mult__inverse__of__nat__commute,axiom,
    ! [Xa3: nat,X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa3))),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa3))) ).

% mult_inverse_of_nat_commute
tff(fact_3167_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3168_mult__inverse__of__int__commute,axiom,
    ! [Xa3: int,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa3))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa3))) ).

% mult_inverse_of_int_commute
tff(fact_3169_mult__inverse__of__int__commute,axiom,
    ! [Xa3: int,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa3))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa3))) ).

% mult_inverse_of_int_commute
tff(fact_3170_mult__inverse__of__int__commute,axiom,
    ! [Xa3: int,X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa3))),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa3))) ).

% mult_inverse_of_int_commute
tff(fact_3171_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_3172_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_3173_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_3174_ceiling__divide__upper,axiom,
    ! [Q4: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P3),Q4)))),Q4))) ) ).

% ceiling_divide_upper
tff(fact_3175_ceiling__divide__upper,axiom,
    ! [Q4: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q4)))),Q4))) ) ).

% ceiling_divide_upper
tff(fact_3176_cpmi,axiom,
    ! [D6: int,P: fun_int_bool,P2: fun_int_bool,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
           => ( pp(aa_int_bool(P,X3))
            <=> pp(aa_int_bool(P2,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B2)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( pp(aa_int_bool(P,X3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa_int_bool(P2,X3))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D6))
                    & pp(aa_int_bool(P2,X5)) )
                | ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D6))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,B2)
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X5))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3177_cppi,axiom,
    ! [D6: int,P: fun_int_bool,P2: fun_int_bool,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
           => ( pp(aa_int_bool(P,X3))
            <=> pp(aa_int_bool(P2,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A2)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( pp(aa_int_bool(P,X3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa_int_bool(P2,X3))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D6))
                    & pp(aa_int_bool(P2,X5)) )
                | ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D6))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,A2)
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X5))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3178_ceiling__divide__lower,axiom,
    ! [Q4: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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),Q4)))),one_one_real)),Q4)),P3)) ) ).

% ceiling_divide_lower
tff(fact_3179_ceiling__divide__lower,axiom,
    ! [Q4: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q4)))),one_one_rat)),Q4)),P3)) ) ).

% ceiling_divide_lower
tff(fact_3180_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_3181_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_3182_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_3183_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_3184_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_3185_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_3186_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_3187_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_3188_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_3189_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_3190_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Z)),aa_real_real(divide_divide_real(Y),W))) ) ) ) ) ).

% frac_less2
tff(fact_3191_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).

% frac_less2
tff(fact_3192_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(X),Z)),aa_real_real(divide_divide_real(Y),W))) ) ) ) ) ).

% frac_less
tff(fact_3193_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).

% frac_less
tff(fact_3194_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(X),Z)),aa_real_real(divide_divide_real(Y),W))) ) ) ) ) ).

% frac_le
tff(fact_3195_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).

% frac_le
tff(fact_3196_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) ) ) ).

% div_positive
tff(fact_3197_div__positive,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))) ) ) ).

% div_positive
tff(fact_3198_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3199_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3200_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_3201_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_3202_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3203_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3204_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_3205_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_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_3206_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_3207_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_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_3208_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_3209_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3210_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_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_3211_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),Y)) ) ) ).

% abs_div_pos
tff(fact_3212_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% abs_div_pos
tff(fact_3213_abs__real__def,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_real_def
tff(fact_3214_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3215_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3216_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3217_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3218_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3219_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3220_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3221_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3222_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3223_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3224_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3225_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3226_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3227_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3228_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3229_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3230_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3231_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3232_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3233_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3234_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3235_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3236_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_3237_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_3238_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_3239_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3240_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3241_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3242_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3243_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3244_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3245_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3246_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3247_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_3248_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_3249_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_3250_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3251_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3252_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3253_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3254_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3255_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3256_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3257_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3258_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_3259_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_3260_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_3261_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_3262_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_3263_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_3264_aset_I2_J,axiom,
    ! [D6: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6))) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6))) ) ) ) ) ) ).

% aset(2)
tff(fact_3265_aset_I1_J,axiom,
    ! [D6: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6))) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6))) ) ) ) ) ) ).

% aset(1)
tff(fact_3266_bset_I2_J,axiom,
    ! [D6: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6))) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                | pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6))) ) ) ) ) ) ).

% bset(2)
tff(fact_3267_bset_I1_J,axiom,
    ! [D6: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6))) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X4))
                & pp(aa_int_bool(Q,X4)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6))) ) ) ) ) ) ).

% bset(1)
tff(fact_3268_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3269_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3270_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3271_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3272_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3273_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3274_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_3275_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_3276_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3277_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(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_3278_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_3279_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_3280_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_3281_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_3282_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3283_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3284_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3285_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_3286_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_3287_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_3288_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3289_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3290_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3291_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3292_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3293_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3294_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3295_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3296_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3297_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3298_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3299_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3300_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3301_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3302_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3303_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_3304_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_3305_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3306_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(minus_minus_real(X),one_one_real)) ).

% square_diff_one_factored
tff(fact_3307_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_3308_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_3309_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_3310_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_3311_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_3312_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          | pp(aa_int_bool(aa_int_fun_int_bool(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_3313_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_3314_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_3315_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_3316_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_3317_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_3318_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_3319_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_3320_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_3321_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_3322_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_3323_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_3324_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_3325_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_3326_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3327_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3328_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(B),A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3329_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3330_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3331_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3332_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3333_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_3334_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_3335_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K),L))) ) ) ).

% div_positive_int
tff(fact_3336_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_3337_zdiv__mono2__neg,axiom,
    ! [A: int,B5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B5)),aa_int_int(divide_divide_int(A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3338_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A5),B)),aa_int_int(divide_divide_int(A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_3339_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(divide_divide_int(I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3340_zdiv__mono2,axiom,
    ! [A: int,B5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B5))) ) ) ) ).

% zdiv_mono2
tff(fact_3341_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A5),B))) ) ) ).

% zdiv_mono1
tff(fact_3342_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A)) ) ).

% ex_gt_or_lt
tff(fact_3343_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_3344_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3345_plusinfinity,axiom,
    ! [D: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int,K2: int] :
            ( pp(aa_int_bool(P2,X3))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
             => ( pp(aa_int_bool(P,X3))
              <=> pp(aa_int_bool(P2,X3)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P2,X_1))
           => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).

% plusinfinity
tff(fact_3346_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int,K2: int] :
            ( pp(aa_int_bool(P1,X3))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
             => ( pp(aa_int_bool(P,X3))
              <=> pp(aa_int_bool(P1,X3)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P1,X_1))
           => ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).

% minusinfinity
tff(fact_3347_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3348_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3349_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_3350_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_3351_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3352_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3353_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3354_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3355_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3356_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3357_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3358_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3359_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3360_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3361_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3362_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3363_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3364_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3365_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3366_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3367_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3368_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3369_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3370_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3371_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3372_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3373_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3374_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3375_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_3376_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_3377_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3378_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3379_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3380_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ) ).

% inverse_less_iff
tff(fact_3381_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ) ).

% inverse_less_iff
tff(fact_3382_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ).

% inverse_le_iff
tff(fact_3383_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ).

% inverse_le_iff
tff(fact_3384_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_3385_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_3386_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_3387_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(B2),N)),aa_int_int(divide_divide_int(A2),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_3388_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_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_3389_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_3390_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R3)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R3),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_3391_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: int,Q5: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R3) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R3)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R3),B5))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3392_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int] :
            ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X4: int] :
              ( pp(aa_int_bool(P,X4))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3393_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: int,Q5: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R3) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R3)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R3))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3394_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R3: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R3),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q4)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3395_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R3: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3396_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int] :
            ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X4: int] :
              ( pp(aa_int_bool(P,X4))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3397_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_3398_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_3399_aset_I7_J,axiom,
    ! [D6: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6))) ) ) ) ).

% aset(7)
tff(fact_3400_aset_I5_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(T,A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6)),T)) ) ) ) ) ).

% aset(5)
tff(fact_3401_aset_I4_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(T,A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6) != T ) ) ) ) ) ).

% aset(4)
tff(fact_3402_aset_I3_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D6) = T ) ) ) ) ) ).

% aset(3)
tff(fact_3403_bset_I7_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(T,B2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6))) ) ) ) ) ).

% bset(7)
tff(fact_3404_bset_I5_J,axiom,
    ! [D6: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,B2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6)),T)) ) ) ) ).

% bset(5)
tff(fact_3405_bset_I4_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(T,B2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6) != T ) ) ) ) ) ).

% bset(4)
tff(fact_3406_bset_I3_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D6))
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D6) = T ) ) ) ) ) ).

% bset(3)
tff(fact_3407_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_3408_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_3409_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3410_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3411_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3412_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3413_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3414_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_3415_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_3416_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_3417_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_3418_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_3419_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_3420_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_3421_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_3422_div__minus1__right,axiom,
    ! [A: int] : aa_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_3423_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_3424_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_3425_div__mult__mult1__if,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)) = zero_zero_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_if
tff(fact_3426_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_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)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_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(divide_divide_int(A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_3427_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(A),B) ).

% div_minus_minus
tff(fact_3428_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_3429_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_3430_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_3431_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_3432_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_3433_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_3434_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: 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,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_3435_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_3436_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_3437_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_3438_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_3439_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_3440_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_3441_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_3442_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_3443_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_3444_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_3445_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_3446_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_3447_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_3448_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_3449_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_3450_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_3451_divide__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(divide_divide_real(X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_3452_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4))) ) ) ).

% div_less_iff_less_mult
tff(fact_3453_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X)))) ).

% real_of_nat_div4
tff(fact_3454_div__nat__eqI,axiom,
    ! [N: nat,Q4: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q4))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_3455_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q4)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3456_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J3) )
               => pp(aa_nat_bool(P,I3)) ) ) ) ) ) ).

% split_div
tff(fact_3457_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_3458_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).

% dividend_less_times_div
tff(fact_3459_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: 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)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_3460_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_3461_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3462_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3463_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3464_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_3465_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3466_add__mult__distrib,axiom,
    ! [M: nat,N: 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),N)),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,N),K)) ).

% add_mult_distrib
tff(fact_3467_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: 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),N)) = 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),N)) ).

% add_mult_distrib2
tff(fact_3468_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),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,N),K)) ).

% diff_mult_distrib
tff(fact_3469_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: 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),N)) = 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),N)) ).

% diff_mult_distrib2
tff(fact_3470_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_3471_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_3472_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3473_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_3474_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q6)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q6))))
            & pp(aa_nat_bool(P,Q6)) ) ) ) ).

% split_div'
tff(fact_3475_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))))) ).

% real_of_nat_div2
tff(fact_3476_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N))
          & pp(aa_nat_bool(P,M4)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less
tff(fact_3477_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N))
         => pp(aa_nat_bool(P,M4)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less
tff(fact_3478_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_3479_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_3480_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_3481_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_3482_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3483_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_3484_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3485_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3486_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_3487_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_3488_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3489_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))) ).

% real_minus_mult_self_le
tff(fact_3490_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(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_3491_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_3492_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = 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),N)),N)) ) ) ) ).

% div_if
tff(fact_3493_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_3494_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_3495_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% one_less_mult
tff(fact_3496_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_3497_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_3498_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y4: real] :
        ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% reals_Archimedean3
tff(fact_3499_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N),X))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X)))) ).

% real_of_int_div4
tff(fact_3500_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_3501_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = 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),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_3502_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = 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),N)),N)) ) ) ) ).

% div_geq
tff(fact_3503_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_3504_nat__mult__distrib,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) ) ) ).

% nat_mult_distrib
tff(fact_3505_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_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_3506_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_3507_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z3)) = 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,Z3))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3508_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N),X))))) ).

% real_of_int_div2
tff(fact_3509_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N),X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_3510_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_3511_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_3512_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3513_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3514_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3515_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_3516_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% div_mult2_eq'
tff(fact_3517_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% div_mult2_eq'
tff(fact_3518_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_3519_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3520_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_3521_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_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(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3522_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3523_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_3524_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_3525_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3526_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3527_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3528_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_3529_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_3530_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_3531_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_3532_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% bits_div_by_1
tff(fact_3533_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_3534_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_3535_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_3536_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_3537_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: 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),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_3538_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_3539_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_3540_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3541_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_3542_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3543_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3544_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3545_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3546_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3547_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3548_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_3549_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_3550_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N) )
      <=> ( 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)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3551_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_3552_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_3553_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)) = 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)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_3554_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)) = 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)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_3555_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3556_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),aa_rat_rat(times_times_rat(Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3557_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3558_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3559_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3560_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3561_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_3562_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_3563_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_3564_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(B2)),uminus5710092332889474511et_nat(A2))) ) ).

% Compl_anti_mono
tff(fact_3565_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B2)))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ).

% Compl_subset_Compl_iff
tff(fact_3566_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_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_3567_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_3568_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_3569_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_3570_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_real(ring_1_of_int_real,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_Ints
tff(fact_3571_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_Code_integer(ring_18347121197199848620nteger,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_Ints
tff(fact_3572_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_rat(ring_1_of_int_rat,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_Ints
tff(fact_3573_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_int(ring_1_of_int_int,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_Ints
tff(fact_3574_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3575_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3576_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3577_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3578_ComplI,axiom,
    ! [C: complex,A2: set_complex] :
      ( ~ member_complex(C,A2)
     => member_complex(C,uminus8566677241136511917omplex(A2)) ) ).

% ComplI
tff(fact_3579_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ member_real(C,A2)
     => member_real(C,uminus612125837232591019t_real(A2)) ) ).

% ComplI
tff(fact_3580_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ member_set_nat(C,A2)
     => member_set_nat(C,uminus613421341184616069et_nat(A2)) ) ).

% ComplI
tff(fact_3581_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(C),A2))
     => pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2))) ) ).

% ComplI
tff(fact_3582_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ member_int(C,A2)
     => member_int(C,uminus1532241313380277803et_int(A2)) ) ).

% ComplI
tff(fact_3583_Compl__iff,axiom,
    ! [C: complex,A2: set_complex] :
      ( member_complex(C,uminus8566677241136511917omplex(A2))
    <=> ~ member_complex(C,A2) ) ).

% Compl_iff
tff(fact_3584_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( member_real(C,uminus612125837232591019t_real(A2))
    <=> ~ member_real(C,A2) ) ).

% Compl_iff
tff(fact_3585_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( member_set_nat(C,uminus613421341184616069et_nat(A2))
    <=> ~ member_set_nat(C,A2) ) ).

% Compl_iff
tff(fact_3586_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
    <=> ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% Compl_iff
tff(fact_3587_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( member_int(C,uminus1532241313380277803et_int(A2))
    <=> ~ member_int(C,A2) ) ).

% Compl_iff
tff(fact_3588_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_3589_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_3590_power__one,axiom,
    ! [N: nat] : aa_nat_real(power_power_real(one_one_real),N) = one_one_real ).

% power_one
tff(fact_3591_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_3592_power__one,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(one_one_complex),N) = one_one_complex ).

% power_one
tff(fact_3593_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,X) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3594_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,X) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3595_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X) = aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3596_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,X) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3597_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,X) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3598_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,X) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3599_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3600_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3601_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3602_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3603_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3604_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3605_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),N) ).

% of_nat_power
tff(fact_3606_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),N) ).

% of_nat_power
tff(fact_3607_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,M)),N) ).

% of_nat_power
tff(fact_3608_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).

% of_nat_power
tff(fact_3609_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),N) ).

% of_nat_power
tff(fact_3610_of__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),N) ).

% of_nat_power
tff(fact_3611_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_3612_power__one__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),one_one_nat) = A ).

% power_one_right
tff(fact_3613_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_3614_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_3615_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(power_power_real(A),M) = aa_nat_real(power_power_real(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3616_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3617_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3618_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_3619_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_3620_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_3621_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_3622_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_3623_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_3624_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_3625_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_3626_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_3627_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_3628_abs__power__minus,axiom,
    ! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) ).

% abs_power_minus
tff(fact_3629_abs__power__minus,axiom,
    ! [A: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N)) = aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),N)) ).

% abs_power_minus
tff(fact_3630_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% abs_power_minus
tff(fact_3631_abs__power__minus,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),N)) ).

% abs_power_minus
tff(fact_3632_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_3633_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_3634_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_3635_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_3636_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_3637_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(gbinomial_real(A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_3638_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_3639_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_3640_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_3641_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,X) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3642_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3643_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3644_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3645_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3646_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) = aa_int_Code_integer(ring_18347121197199848620nteger,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3647_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: 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,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3648_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3649_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3650_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3651_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(power_power_int(Z),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)),N) ).

% of_int_power
tff(fact_3652_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_3653_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_3654_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),N)) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_3655_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),N)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).

% of_int_power
tff(fact_3656_left__minus__one__mult__self,axiom,
    ! [N: 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)),N)),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)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3657_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3658_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3659_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3660_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3661_minus__one__mult__self,axiom,
    ! [N: 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)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_3662_minus__one__mult__self,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).

% minus_one_mult_self
tff(fact_3663_minus__one__mult__self,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).

% minus_one_mult_self
tff(fact_3664_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_3665_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_3666_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),X)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3667_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3668_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3669_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_3670_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3671_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3672_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(power_power_real(A),N) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3673_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3674_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_3675_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3676_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3677_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3678_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3679_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3680_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3681_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3682_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3683_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3684_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3685_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3686_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3687_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3688_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3689_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3690_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3691_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3692_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3693_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3694_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3695_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3696_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3697_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3698_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3699_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),X)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3700_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3701_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3702_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_3703_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_3704_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_3705_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_3706_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_3707_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3708_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3709_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3710_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3711_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3712_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3713_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3714_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3715_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3716_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3717_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3718_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3719_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3720_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3721_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3722_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3723_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3724_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3725_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3726_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3727_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3728_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3729_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3730_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_3731_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3732_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3733_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3734_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3735_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3736_ComplD,axiom,
    ! [C: complex,A2: set_complex] :
      ( member_complex(C,uminus8566677241136511917omplex(A2))
     => ~ member_complex(C,A2) ) ).

% ComplD
tff(fact_3737_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( member_real(C,uminus612125837232591019t_real(A2))
     => ~ member_real(C,A2) ) ).

% ComplD
tff(fact_3738_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( member_set_nat(C,uminus613421341184616069et_nat(A2))
     => ~ member_set_nat(C,A2) ) ).

% ComplD
tff(fact_3739_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% ComplD
tff(fact_3740_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( member_int(C,uminus1532241313380277803et_int(A2))
     => ~ member_int(C,A2) ) ).

% ComplD
tff(fact_3741_Ints__power,axiom,
    ! [A: int,N: nat] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_nat_int(power_power_int(A),N),ring_1_Ints_int) ) ).

% Ints_power
tff(fact_3742_Ints__power,axiom,
    ! [A: real,N: nat] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_nat_real(power_power_real(A),N),ring_1_Ints_real) ) ).

% Ints_power
tff(fact_3743_Ints__power,axiom,
    ! [A: complex,N: nat] :
      ( member_complex(A,ring_1_Ints_complex)
     => member_complex(aa_nat_complex(power_power_complex(A),N),ring_1_Ints_complex) ) ).

% Ints_power
tff(fact_3744_of__nat__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,N),K)) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),K) ).

% of_nat_gbinomial
tff(fact_3745_of__nat__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,N),K)) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) ).

% of_nat_gbinomial
tff(fact_3746_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(gbinomial_real(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3747_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3748_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_3749_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_3750_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_3751_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_3752_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_3753_Ints__0,axiom,
    member_complex(zero_zero_complex,ring_1_Ints_complex) ).

% Ints_0
tff(fact_3754_Ints__0,axiom,
    member_real(zero_zero_real,ring_1_Ints_real) ).

% Ints_0
tff(fact_3755_Ints__0,axiom,
    member_rat(zero_zero_rat,ring_1_Ints_rat) ).

% Ints_0
tff(fact_3756_Ints__0,axiom,
    member_int(zero_zero_int,ring_1_Ints_int) ).

% Ints_0
tff(fact_3757_Ints__mult,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(times_times_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_mult
tff(fact_3758_Ints__mult,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_mult
tff(fact_3759_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(times_times_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_3760_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_mult
tff(fact_3761_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_3762_Ints__1,axiom,
    member_int(one_one_int,ring_1_Ints_int) ).

% Ints_1
tff(fact_3763_Ints__1,axiom,
    member_real(one_one_real,ring_1_Ints_real) ).

% Ints_1
tff(fact_3764_Ints__1,axiom,
    member_complex(one_one_complex,ring_1_Ints_complex) ).

% Ints_1
tff(fact_3765_Ints__add,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(plus_plus_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_add
tff(fact_3766_Ints__add,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_add
tff(fact_3767_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_3768_Ints__add,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_add
tff(fact_3769_power__inverse,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,A)),N) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(A),N)) ).

% power_inverse
tff(fact_3770_power__inverse,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),N) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),N)) ).

% power_inverse
tff(fact_3771_power__inverse,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),N) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),N)) ).

% power_inverse
tff(fact_3772_Ints__diff,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(minus_minus_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_diff
tff(fact_3773_Ints__diff,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(minus_minus_real(A),B),ring_1_Ints_real) ) ) ).

% Ints_diff
tff(fact_3774_Ints__diff,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(minus_minus_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_diff
tff(fact_3775_Ints__diff,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_diff
tff(fact_3776_Ints__minus,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_int_int(uminus_uminus_int,A),ring_1_Ints_int) ) ).

% Ints_minus
tff(fact_3777_Ints__minus,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_real_real(uminus_uminus_real,A),ring_1_Ints_real) ) ).

% Ints_minus
tff(fact_3778_Ints__minus,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => member_complex(aa_complex_complex(uminus1482373934393186551omplex,A),ring_1_Ints_complex) ) ).

% Ints_minus
tff(fact_3779_Ints__minus,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),ring_11222124179247155820nteger) ) ).

% Ints_minus
tff(fact_3780_Ints__minus,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(aa_rat_rat(uminus_uminus_rat,A),ring_1_Ints_rat) ) ).

% Ints_minus
tff(fact_3781_minus__in__Ints__iff,axiom,
    ! [X: int] :
      ( member_int(aa_int_int(uminus_uminus_int,X),ring_1_Ints_int)
    <=> member_int(X,ring_1_Ints_int) ) ).

% minus_in_Ints_iff
tff(fact_3782_minus__in__Ints__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(uminus_uminus_real,X),ring_1_Ints_real)
    <=> member_real(X,ring_1_Ints_real) ) ).

% minus_in_Ints_iff
tff(fact_3783_minus__in__Ints__iff,axiom,
    ! [X: complex] :
      ( member_complex(aa_complex_complex(uminus1482373934393186551omplex,X),ring_1_Ints_complex)
    <=> member_complex(X,ring_1_Ints_complex) ) ).

% minus_in_Ints_iff
tff(fact_3784_minus__in__Ints__iff,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),ring_11222124179247155820nteger)
    <=> member_Code_integer(X,ring_11222124179247155820nteger) ) ).

% minus_in_Ints_iff
tff(fact_3785_minus__in__Ints__iff,axiom,
    ! [X: rat] :
      ( member_rat(aa_rat_rat(uminus_uminus_rat,X),ring_1_Ints_rat)
    <=> member_rat(X,ring_1_Ints_rat) ) ).

% minus_in_Ints_iff
tff(fact_3786_Ints__of__nat,axiom,
    ! [N: nat] : member_complex(aa_nat_complex(semiri8010041392384452111omplex,N),ring_1_Ints_complex) ).

% Ints_of_nat
tff(fact_3787_Ints__of__nat,axiom,
    ! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),ring_1_Ints_int) ).

% Ints_of_nat
tff(fact_3788_Ints__of__nat,axiom,
    ! [N: nat] : member_real(aa_nat_real(semiri5074537144036343181t_real,N),ring_1_Ints_real) ).

% Ints_of_nat
tff(fact_3789_Ints__of__nat,axiom,
    ! [N: nat] : member_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N),ring_11222124179247155820nteger) ).

% Ints_of_nat
tff(fact_3790_Ints__of__nat,axiom,
    ! [N: nat] : member_rat(aa_nat_rat(semiri681578069525770553at_rat,N),ring_1_Ints_rat) ).

% Ints_of_nat
tff(fact_3791_Ints__abs,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_int_int(abs_abs_int,A),ring_1_Ints_int) ) ).

% Ints_abs
tff(fact_3792_Ints__abs,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(abs_abs_Code_integer(A),ring_11222124179247155820nteger) ) ).

% Ints_abs
tff(fact_3793_Ints__abs,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(aa_rat_rat(abs_abs_rat,A),ring_1_Ints_rat) ) ).

% Ints_abs
tff(fact_3794_Ints__abs,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_real_real(abs_abs_real,A),ring_1_Ints_real) ) ).

% Ints_abs
tff(fact_3795_Ints__cases,axiom,
    ! [Q4: int] :
      ( member_int(Q4,ring_1_Ints_int)
     => ~ ! [Z2: int] : Q4 != aa_int_int(ring_1_of_int_int,Z2) ) ).

% Ints_cases
tff(fact_3796_Ints__cases,axiom,
    ! [Q4: real] :
      ( member_real(Q4,ring_1_Ints_real)
     => ~ ! [Z2: int] : Q4 != aa_int_real(ring_1_of_int_real,Z2) ) ).

% Ints_cases
tff(fact_3797_Ints__cases,axiom,
    ! [Q4: code_integer] :
      ( member_Code_integer(Q4,ring_11222124179247155820nteger)
     => ~ ! [Z2: int] : Q4 != aa_int_Code_integer(ring_18347121197199848620nteger,Z2) ) ).

% Ints_cases
tff(fact_3798_Ints__cases,axiom,
    ! [Q4: rat] :
      ( member_rat(Q4,ring_1_Ints_rat)
     => ~ ! [Z2: int] : Q4 != aa_int_rat(ring_1_of_int_rat,Z2) ) ).

% Ints_cases
tff(fact_3799_Ints__cases,axiom,
    ! [Q4: complex] :
      ( member_complex(Q4,ring_1_Ints_complex)
     => ~ ! [Z2: int] : Q4 != aa_int_complex(ring_17405671764205052669omplex,Z2) ) ).

% Ints_cases
tff(fact_3800_Ints__induct,axiom,
    ! [Q4: int,P: fun_int_bool] :
      ( member_int(Q4,ring_1_Ints_int)
     => ( ! [Z2: int] : pp(aa_int_bool(P,aa_int_int(ring_1_of_int_int,Z2)))
       => pp(aa_int_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_3801_Ints__induct,axiom,
    ! [Q4: real,P: fun_real_bool] :
      ( member_real(Q4,ring_1_Ints_real)
     => ( ! [Z2: int] : pp(aa_real_bool(P,aa_int_real(ring_1_of_int_real,Z2)))
       => pp(aa_real_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_3802_Ints__induct,axiom,
    ! [Q4: code_integer,P: fun_Co8822979462182099027r_bool] :
      ( member_Code_integer(Q4,ring_11222124179247155820nteger)
     => ( ! [Z2: int] : pp(aa_Code_integer_bool(P,aa_int_Code_integer(ring_18347121197199848620nteger,Z2)))
       => pp(aa_Code_integer_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_3803_Ints__induct,axiom,
    ! [Q4: rat,P: fun_rat_bool] :
      ( member_rat(Q4,ring_1_Ints_rat)
     => ( ! [Z2: int] : pp(aa_rat_bool(P,aa_int_rat(ring_1_of_int_rat,Z2)))
       => pp(aa_rat_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_3804_Ints__induct,axiom,
    ! [Q4: complex,P: fun_complex_bool] :
      ( member_complex(Q4,ring_1_Ints_complex)
     => ( ! [Z2: int] : pp(aa_complex_bool(P,aa_int_complex(ring_17405671764205052669omplex,Z2)))
       => pp(aa_complex_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_3805_Ints__of__int,axiom,
    ! [Z: int] : member_int(aa_int_int(ring_1_of_int_int,Z),ring_1_Ints_int) ).

% Ints_of_int
tff(fact_3806_Ints__of__int,axiom,
    ! [Z: int] : member_real(aa_int_real(ring_1_of_int_real,Z),ring_1_Ints_real) ).

% Ints_of_int
tff(fact_3807_Ints__of__int,axiom,
    ! [Z: int] : member_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Z),ring_11222124179247155820nteger) ).

% Ints_of_int
tff(fact_3808_Ints__of__int,axiom,
    ! [Z: int] : member_rat(aa_int_rat(ring_1_of_int_rat,Z),ring_1_Ints_rat) ).

% Ints_of_int
tff(fact_3809_Ints__of__int,axiom,
    ! [Z: int] : member_complex(aa_int_complex(ring_17405671764205052669omplex,Z),ring_1_Ints_complex) ).

% Ints_of_int
tff(fact_3810_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),N)) ).

% gbinomial_index_swap
tff(fact_3811_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),N)) ).

% gbinomial_index_swap
tff(fact_3812_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),N)) ).

% gbinomial_index_swap
tff(fact_3813_gbinomial__negated__upper,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_3814_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_3815_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_3816_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_3817_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_3818_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_3819_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ).

% power_mono
tff(fact_3820_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono
tff(fact_3821_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).

% power_mono
tff(fact_3822_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono
tff(fact_3823_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_le_power
tff(fact_3824_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_power
tff(fact_3825_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_le_power
tff(fact_3826_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_power
tff(fact_3827_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_less_power
tff(fact_3828_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_less_power
tff(fact_3829_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_less_power
tff(fact_3830_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_less_power
tff(fact_3831_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(A),N))) ) ).

% one_le_power
tff(fact_3832_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% one_le_power
tff(fact_3833_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% one_le_power
tff(fact_3834_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).

% one_le_power
tff(fact_3835_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_3836_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),N)),aa_nat_real(power_power_real(Y),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_3837_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_3838_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_3839_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_3840_power__one__over,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),N) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),N)) ).

% power_one_over
tff(fact_3841_power__one__over,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(aa_real_real(divide_divide_real(one_one_real),A)),N) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(A),N)) ).

% power_one_over
tff(fact_3842_power__one__over,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),N) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),N)) ).

% power_one_over
tff(fact_3843_power__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_Suc
tff(fact_3844_power__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)) ).

% power_Suc
tff(fact_3845_power__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).

% power_Suc
tff(fact_3846_power__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_3847_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_3848_power__Suc2,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) ).

% power_Suc2
tff(fact_3849_power__Suc2,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),N)),A) ).

% power_Suc2
tff(fact_3850_power__Suc2,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) ).

% power_Suc2
tff(fact_3851_power__Suc2,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).

% power_Suc2
tff(fact_3852_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_3853_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3854_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3855_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3856_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_3857_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3858_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3859_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3860_power__add,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_3861_power__add,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ).

% power_add
tff(fact_3862_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_3863_power__add,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_3864_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_3865_power__mult__power__inverse__commute,axiom,
    ! [X: real,M: nat,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),N)),aa_nat_real(power_power_real(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_3866_power__mult__power__inverse__commute,axiom,
    ! [X: complex,M: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),N)),aa_nat_complex(power_power_complex(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_3867_power__mult__power__inverse__commute,axiom,
    ! [X: rat,M: nat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),N)),aa_nat_rat(power_power_rat(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_3868_power__mult__inverse__distrib,axiom,
    ! [X: real,M: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_real_real(inverse_inverse_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,X)),aa_nat_real(power_power_real(X),M)) ).

% power_mult_inverse_distrib
tff(fact_3869_power__mult__inverse__distrib,axiom,
    ! [X: complex,M: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(invers8013647133539491842omplex,X)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,X)),aa_nat_complex(power_power_complex(X),M)) ).

% power_mult_inverse_distrib
tff(fact_3870_power__mult__inverse__distrib,axiom,
    ! [X: rat,M: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(inverse_inverse_rat,X)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,X)),aa_nat_rat(power_power_rat(X),M)) ).

% power_mult_inverse_distrib
tff(fact_3871_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( ( aa_complex_complex(plus_plus_complex(A),A) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3872_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3873_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3874_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3875_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_3876_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_3877_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_3878_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_3879_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_3880_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_3881_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_3882_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_3883_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_3884_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_3885_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_3886_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_3887_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_3888_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3889_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3890_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3891_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3892_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_3893_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_3894_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_3895_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_3896_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),aa_nat_real(power_power_real(B),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3897_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3898_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3899_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3900_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_nat_real(power_power_real(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3901_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3902_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3903_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3904_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3905_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3906_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_3907_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3908_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3909_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3910_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_gt1_lemma
tff(fact_3911_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(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),N)))) ) ).

% power_gt1_lemma
tff(fact_3912_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_3913_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_3914_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_3915_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_3916_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_3917_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3918_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3919_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3920_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3921_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).

% power_minus
tff(fact_3922_power__minus,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(power_power_real(A),N)) ).

% power_minus
tff(fact_3923_power__minus,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(A),N)) ).

% power_minus
tff(fact_3924_power__minus,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_minus
tff(fact_3925_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_3926_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N))) ).

% zero_le_power_abs
tff(fact_3927_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_3928_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N))) ).

% zero_le_power_abs
tff(fact_3929_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).

% zero_le_power_abs
tff(fact_3930_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3931_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3932_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3933_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3934_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_3935_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_3936_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_strict_increasing
tff(fact_3937_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_3938_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(A),N5))) ) ) ).

% power_increasing
tff(fact_3939_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N5))) ) ) ).

% power_increasing
tff(fact_3940_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_increasing
tff(fact_3941_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_increasing
tff(fact_3942_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_3943_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_3944_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_3945_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_3946_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_3947_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A)),A) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_3948_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_3949_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_3950_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_3951_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_3952_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(gbinomial_real(A),K)) ).

% Suc_times_gbinomial
tff(fact_3953_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_3954_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_3955_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_3956_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_3957_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),M)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3958_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3959_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N))),aa_nat_real(power_power_real(A),N))) ) ) ).

% power_Suc_less
tff(fact_3960_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% power_Suc_less
tff(fact_3961_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_3962_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_3963_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_3964_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_3965_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_3966_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_3967_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_3968_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_3969_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_3970_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_3971_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N5)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3972_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N5)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3973_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3974_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N5)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3975_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N5)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_decreasing
tff(fact_3976_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N5)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_decreasing
tff(fact_3977_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_decreasing
tff(fact_3978_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N5)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_3979_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3980_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3981_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3982_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_3983_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( ( aa_nat_real(power_power_real(A),N) = aa_nat_real(power_power_real(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3984_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3985_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3986_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3987_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),N) = aa_nat_real(power_power_real(B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3988_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3989_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3990_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3991_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(power_power_real(A),N))) ) ) ).

% self_le_power
tff(fact_3992_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% self_le_power
tff(fact_3993_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% self_le_power
tff(fact_3994_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_3995_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),N))) ) ) ).

% one_less_power
tff(fact_3996_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% one_less_power
tff(fact_3997_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% one_less_power
tff(fact_3998_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_3999_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_4000_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ) ) ) ).

% power_diff
tff(fact_4001_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_4002_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_4003_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_4004_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% Ints_odd_less_0
tff(fact_4005_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% Ints_odd_less_0
tff(fact_4006_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Ints_odd_less_0
tff(fact_4007_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_4008_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( ( X != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_4009_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( ( X != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_4010_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( ( X != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_4011_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer))
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_4012_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_4013_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat))
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_4014_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int))
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_4015_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y,ring_11222124179247155820nteger)
       => ( ( X = Y )
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Y))),one_one_Code_integer)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_4016_Ints__eq__abs__less1,axiom,
    ! [X: real,Y: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( member_real(Y,ring_1_Ints_real)
       => ( ( X = Y )
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y))),one_one_real)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_4017_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( member_rat(Y,ring_1_Ints_rat)
       => ( ( X = Y )
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),Y))),one_one_rat)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_4018_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( member_int(Y,ring_1_Ints_int)
       => ( ( X = Y )
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_4019_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_4020_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_4021_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_4022_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_4023_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(gbinomial_real(A),K)) ).

% gbinomial_factors
tff(fact_4024_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_4025_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_4026_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_4027_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_4028_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_4029_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4030_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_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,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4031_power__diff__conv__inverse,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),N)),aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4032_power__diff__conv__inverse,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( X != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_complex(power_power_complex(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4033_power__diff__conv__inverse,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( X != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_rat(power_power_rat(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4034_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = 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_4035_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_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_4036_power__eq__if,axiom,
    ! [M: nat,P3: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = aa_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_4037_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = 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_4038_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = 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_4039_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_4040_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_4041_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_4042_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_4043_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(power_power_real(A),N) ) ) ).

% power_minus_mult
tff(fact_4044_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_4045_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_4046_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_4047_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( 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,N),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,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4048_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_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,N),K)) = aa_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,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4049_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4050_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4051_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4052_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = A )
    <=> ( member_real(aa_real_real(minus_minus_real(X),A),ring_1_Ints_real)
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) ) ) ).

% frac_unique_iff
tff(fact_4053_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = A )
    <=> ( member_rat(aa_rat_rat(minus_minus_rat(X),A),ring_1_Ints_rat)
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) ) ) ).

% frac_unique_iff
tff(fact_4054_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_4055_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_4056_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_4057_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_4058_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_4059_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_4060_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_4061_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_4062_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_4063_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_4064_frac__neg,axiom,
    ! [X: real] :
      ( ( member_real(X,ring_1_Ints_real)
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = zero_zero_real ) )
      & ( ~ member_real(X,ring_1_Ints_real)
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(one_one_real),aa_real_real(archim2898591450579166408c_real,X)) ) ) ) ).

% frac_neg
tff(fact_4065_frac__neg,axiom,
    ! [X: rat] :
      ( ( member_rat(X,ring_1_Ints_rat)
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = zero_zero_rat ) )
      & ( ~ member_rat(X,ring_1_Ints_rat)
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,X)) ) ) ) ).

% frac_neg
tff(fact_4066_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_4067_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_4068_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_4069_frac__in__Ints__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(archim2898591450579166408c_real,X),ring_1_Ints_real)
    <=> member_real(X,ring_1_Ints_real) ) ).

% frac_in_Ints_iff
tff(fact_4070_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_4071_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_4072_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_4073_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_4074_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_4075_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_4076_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_4077_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_4078_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_4079_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_4080_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% frac_eq_0_iff
tff(fact_4081_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = zero_zero_rat )
    <=> member_rat(X,ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_4082_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X)))
    <=> ~ member_real(X,ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_4083_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)))
    <=> ~ member_rat(X,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_4084_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_nat_int(semiri1314217659103216013at_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_4085_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_nat_real(semiri5074537144036343181t_real,X)),N) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_4086_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_4087_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_4088_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_4089_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_int_real(ring_1_of_int_real,X)),N) = aa_int_real(ring_1_of_int_real,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_4090_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_int_Code_integer(ring_18347121197199848620nteger,X)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_4091_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_int_rat(ring_1_of_int_rat,X)),N) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_4092_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_int_complex(ring_17405671764205052669omplex,X)),N) = aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_4093_arctan__minus,axiom,
    ! [X: real] : aa_real_real(arctan,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arctan,X)) ).

% arctan_minus
tff(fact_4094_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),N))) ) ).

% pochhammer_pos
tff(fact_4095_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_pos
tff(fact_4096_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_pos
tff(fact_4097_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_pos
tff(fact_4098_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_4099_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),M) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4100_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4101_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4102_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4103_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(power_power_real(X),N2))) ) ).

% real_arch_pow
tff(fact_4104_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_4105_frac__ge__0,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_ge_0
tff(fact_4106_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_4107_frac__lt__1,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).

% frac_lt_1
tff(fact_4108_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_4109_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_4110_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),N))) ) ).

% pochhammer_nonneg
tff(fact_4111_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_nonneg
tff(fact_4112_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_nonneg
tff(fact_4113_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_nonneg
tff(fact_4114_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_4115_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

% nat_one_le_power
tff(fact_4116_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_4117_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_4118_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_4119_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_4120_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_4121_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_4122_pochhammer__rec,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N)) ).

% pochhammer_rec
tff(fact_4123_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_4124_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_4125_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).

% pochhammer_rec
tff(fact_4126_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ).

% pochhammer_rec
tff(fact_4127_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).

% pochhammer_rec'
tff(fact_4128_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(comm_s7457072308508201937r_real(Z),N)) ).

% pochhammer_rec'
tff(fact_4129_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_4130_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)) ).

% pochhammer_rec'
tff(fact_4131_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_4132_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_4133_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(A),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_4134_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_4135_pochhammer__Suc,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).

% pochhammer_Suc
tff(fact_4136_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_4137_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4138_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4139_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4140_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4141_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4142_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4143_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4144_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4145_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4146_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4147_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_4148_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_4149_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_4150_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4151_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4152_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4153_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4154_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4155_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_4156_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),N)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_4157_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_4158_pochhammer__product_H,axiom,
    ! [Z: code_integer,N: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),M)) ).

% pochhammer_product'
tff(fact_4159_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).

% pochhammer_product'
tff(fact_4160_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_4161_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% frac_eq
tff(fact_4162_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% frac_eq
tff(fact_4163_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_4164_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_4165_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4166_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),M)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4167_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4168_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4169_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4170_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_4171_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_4172_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_4173_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R2),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_4174_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_4175_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)),one_one_real)),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_4176_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4177_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
            & ( aa_nat_real(power_power_real(X3),N) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(power_power_real(Y4),N) = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_4178_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [R4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R4))
            & ( aa_nat_real(power_power_real(R4),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_4179_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R4))
          & ( aa_nat_real(power_power_real(R4),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_4180_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4181_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_4182_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(power_power_real(X),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_4183_arsinh__minus__real,axiom,
    ! [X: real] : aa_real_real(arsinh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arsinh_real,X)) ).

% arsinh_minus_real
tff(fact_4184_powr__0,axiom,
    ! [Z: real] : aa_real_real(powr_real(zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_4185_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_4186_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_4187_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(powr_real(X),zero_zero_real) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(powr_real(X),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_4188_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel_iff
tff(fact_4189_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(powr_real(A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_4190_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(powr_real(X),one_one_real) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_4191_powr__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(powr_real(X),one_one_real) = X ) ) ).

% powr_one
tff(fact_4192_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_4193_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_4194_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B))) ) ) ).

% powr_less_mono
tff(fact_4195_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel
tff(fact_4196_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B))) ) ) ).

% powr_mono
tff(fact_4197_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( ( aa_real_real(powr_real(A),X) = aa_real_real(powr_real(A),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_4198_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(powr_real(X),Y))) ) ) ).

% gr_one_powr
tff(fact_4199_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(powr_real(X),A))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_4200_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y),B))) ) ) ) ) ).

% powr_mono_both
tff(fact_4201_powr__le1,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X),A)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_4202_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_4203_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(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_4204_powr__add,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(powr_real(X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)) ).

% powr_add
tff(fact_4205_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(powr_real(W),aa_real_real(minus_minus_real(Z1),Z22)) = aa_real_real(divide_divide_real(aa_real_real(powr_real(W),Z1)),aa_real_real(powr_real(W),Z22)) ).

% powr_diff
tff(fact_4206_powr__minus,axiom,
    ! [X: real,A: real] : aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(X),A)) ).

% powr_minus
tff(fact_4207_powr__minus__divide,axiom,
    ! [X: real,A: real] : aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,A)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(powr_real(X),A)) ).

% powr_minus_divide
tff(fact_4208_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_4209_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(powr_real(X),Y)) = aa_real_real(powr_real(X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_4210_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(power_power_real(X),aa_int_nat(nat2,N)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_4211_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(B),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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_4212_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_4213_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_4214_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4215_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_4216_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_4217_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_4218_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_4219_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_4220_gbinomial__pochhammer,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_4221_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_4222_gbinomial__pochhammer,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_4223_of__nat__fact,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_int(semiri1406184849735516958ct_int,N) ).

% of_nat_fact
tff(fact_4224_of__nat__fact,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_Code_integer(semiri3624122377584611663nteger,N) ).

% of_nat_fact
tff(fact_4225_of__nat__fact,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_rat(semiri773545260158071498ct_rat,N) ).

% of_nat_fact
tff(fact_4226_of__nat__fact,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% of_nat_fact
tff(fact_4227_of__nat__fact,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_real(semiri2265585572941072030t_real,N) ).

% of_nat_fact
tff(fact_4228_greaterThanLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or8625682525731655386et_nat(L,U))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(L),I))
        & pp(aa_set_nat_bool(ord_less_set_nat(I),U)) ) ) ).

% greaterThanLessThan_iff
tff(fact_4229_greaterThanLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( member_rat(I,set_or5199638295745620268an_rat(L,U))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),U)) ) ) ).

% greaterThanLessThan_iff
tff(fact_4230_greaterThanLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or2392100141987894638an_num(L,U))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),U)) ) ) ).

% greaterThanLessThan_iff
tff(fact_4231_greaterThanLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( member_int(I,set_or5832277885323065728an_int(L,U))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),U)) ) ) ).

% greaterThanLessThan_iff
tff(fact_4232_greaterThanLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( member_real(I,set_or1633881224788618240n_real(L,U))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),U)) ) ) ).

% greaterThanLessThan_iff
tff(fact_4233_greaterThanLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),U)) ) ) ).

% greaterThanLessThan_iff
tff(fact_4234_of__int__fact,axiom,
    ! [N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(semiri1406184849735516958ct_int,N)) = aa_nat_Code_integer(semiri3624122377584611663nteger,N) ).

% of_int_fact
tff(fact_4235_of__int__fact,axiom,
    ! [N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1406184849735516958ct_int,N)) = aa_nat_rat(semiri773545260158071498ct_rat,N) ).

% of_int_fact
tff(fact_4236_of__int__fact,axiom,
    ! [N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(semiri1406184849735516958ct_int,N)) = aa_nat_complex(semiri5044797733671781792omplex,N) ).

% of_int_fact
tff(fact_4237_of__int__fact,axiom,
    ! [N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1406184849735516958ct_int,N)) = aa_nat_real(semiri2265585572941072030t_real,N) ).

% of_int_fact
tff(fact_4238_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_4239_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_4240_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_4241_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_4242_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_4243_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_4244_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_4245_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_4246_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_4247_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_4248_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_4249_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_4250_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_4251_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_4252_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_4253_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_4254_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_4255_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4256_log__one,axiom,
    ! [A: real] : aa_real_real(log2(A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_4257_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,X) = one_one_real )
    <=> ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_4258_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_4259_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_4260_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_4261_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_4262_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_4263_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_4264_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_4265_fact__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_Suc
tff(fact_4266_fact__Suc,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% fact_Suc
tff(fact_4267_fact__Suc,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_Suc
tff(fact_4268_fact__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_Suc
tff(fact_4269_fact__Suc,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_Suc
tff(fact_4270_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_4271_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_4272_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_4273_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_4274_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_4275_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(log2(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_4276_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_4277_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(log2(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_4278_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_4279_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(A),X)),aa_real_real(log2(A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_4280_log__eq__one,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log2(A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_4281_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_less_exp_iff
tff(fact_4282_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_4283_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_4284_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% one_le_exp_iff
tff(fact_4285_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_4286_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_4287_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_4288_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_4289_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_4290_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_4291_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_4292_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_4293_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(log2(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_4294_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_4295_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(log2(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_4296_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_4297_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(A),X)),aa_real_real(log2(A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_4298_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(powr_real(A),aa_real_real(log2(A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_4299_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log2(A),aa_real_real(powr_real(A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_4300_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4301_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log2(A),aa_nat_real(power_power_real(A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_4302_log__ln,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = aa_real_real(log2(aa_real_real(exp_real,one_one_real)),X) ).

% log_ln
tff(fact_4303_real__root__minus,axiom,
    ! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_minus
tff(fact_4304_exp__not__eq__zero,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_4305_exp__not__eq__zero,axiom,
    ! [X: real] : aa_real_real(exp_real,X) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_4306_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).

% fact_nonzero
tff(fact_4307_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_4308_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_4309_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) != zero_zero_real ).

% fact_nonzero
tff(fact_4310_real__root__inverse,axiom,
    ! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_inverse
tff(fact_4311_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_4312_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_4313_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_zero
tff(fact_4314_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_zero
tff(fact_4315_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_zero
tff(fact_4316_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_zero
tff(fact_4317_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat)) ).

% fact_not_neg
tff(fact_4318_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).

% fact_not_neg
tff(fact_4319_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).

% fact_not_neg
tff(fact_4320_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N)),zero_zero_real)) ).

% fact_not_neg
tff(fact_4321_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_gt_zero
tff(fact_4322_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_gt_zero
tff(fact_4323_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_gt_zero
tff(fact_4324_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_gt_zero
tff(fact_4325_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% mult_exp_exp
tff(fact_4326_mult__exp__exp,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% mult_exp_exp
tff(fact_4327_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_4328_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_4329_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_1
tff(fact_4330_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_1
tff(fact_4331_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_1
tff(fact_4332_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_1
tff(fact_4333_exp__diff,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_4334_exp__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(exp_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_4335_powr__def,axiom,
    ! [X: real,A: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(powr_real(X),A) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(powr_real(X),A) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,X))) ) ) ) ).

% powr_def
tff(fact_4336_exp__minus,axiom,
    ! [X: real] : aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)) ).

% exp_minus
tff(fact_4337_exp__minus,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(exp_complex,X)) ).

% exp_minus
tff(fact_4338_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),N) ).

% pochhammer_fact
tff(fact_4339_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N) ).

% pochhammer_fact
tff(fact_4340_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),N) ).

% pochhammer_fact
tff(fact_4341_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N) ).

% pochhammer_fact
tff(fact_4342_pochhammer__fact,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = aa_nat_real(comm_s7457072308508201937r_real(one_one_real),N) ).

% pochhammer_fact
tff(fact_4343_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( ( B != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( aa_real_real(log2(A),X) = aa_real_real(aa_rea7340605660300375871l_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(log2(B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_4344_ln__root,axiom,
    ! [N: nat,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_4345_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( aa_real_real(log2(B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(divide_divide_real(aa_real_real(log2(B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_4346_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(log2(aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(B),X)) ) ) ) ).

% log_base_root
tff(fact_4347_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_4348_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_4349_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_gt_zero
tff(fact_4350_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_ge_zero
tff(fact_4351_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_4352_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_4353_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_4354_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(X),K)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).

% real_root_power
tff(fact_4355_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).

% exp_gt_one
tff(fact_4356_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_4357_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ).

% exp_ge_add_one_self
tff(fact_4358_exp__minus__inverse,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) = one_one_real ).

% exp_minus_inverse
tff(fact_4359_exp__minus__inverse,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) = one_one_complex ).

% exp_minus_inverse
tff(fact_4360_exp__of__nat2__mult,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(X),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),N) ).

% exp_of_nat2_mult
tff(fact_4361_exp__of__nat2__mult,axiom,
    ! [X: real,N: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(power_power_real(aa_real_real(exp_real,X)),N) ).

% exp_of_nat2_mult
tff(fact_4362_exp__of__nat__mult,axiom,
    ! [N: nat,X: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),X)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),N) ).

% exp_of_nat_mult
tff(fact_4363_exp__of__nat__mult,axiom,
    ! [N: nat,X: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) = aa_nat_real(power_power_real(aa_real_real(exp_real,X)),N) ).

% exp_of_nat_mult
tff(fact_4364_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ) ).

% fact_less_mono
tff(fact_4365_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).

% fact_less_mono
tff(fact_4366_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono
tff(fact_4367_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ) ).

% fact_less_mono
tff(fact_4368_fact__le__power,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_4369_fact__le__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_4370_fact__le__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_4371_fact__le__power,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_4372_fact__le__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_4373_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or5199638295745620268an_rat(A,B),set_or5199638295745620268an_rat(C,D))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4374_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1633881224788618240n_real(A,B),set_or1633881224788618240n_real(C,D))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4375_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_4376_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_4377_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N5),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4378_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_4379_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(minus_minus_real(X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_4380_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log2(B),X) = aa_real_real(divide_divide_real(aa_real_real(log2(A),X)),aa_real_real(log2(A),B)) ) ) ) ).

% log_base_change
tff(fact_4381_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_4382_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_4383_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_4384_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(minus_minus_real(Y),one_one_real)))
          & ( aa_real_real(exp_real,X3) = Y ) ) ) ).

% lemma_exp_total
tff(fact_4385_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(log2(B),X))) ) ) ) ).

% powr_less_iff
tff(fact_4386_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(powr_real(B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(B),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_4387_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(powr_real(B),Y))) ) ) ) ).

% log_less_iff
tff(fact_4388_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(log2(B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_4389_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(power_power_real(B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(log2(B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_4390_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(B),M))) ) ) ).

% less_log_of_power
tff(fact_4391_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_4392_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or5199638295745620268an_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4393_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1633881224788618240n_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4394_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_4395_real__root__strict__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4396_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N5),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_4397_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_4398_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4399_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4400_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(minus_minus_real(X),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_4401_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_4402_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(log2(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log2(A),X)),aa_real_real(log2(A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_4403_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(log2(B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_4404_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(powr_real(B),Y))) ) ) ) ).

% log_le_iff
tff(fact_4405_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(powr_real(B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(B),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_4406_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(log2(B),X))) ) ) ) ).

% powr_le_iff
tff(fact_4407_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(log2(A),aa_real_real(divide_divide_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(log2(A),X)),aa_real_real(log2(A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_4408_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(B),M))) ) ) ).

% le_log_of_power
tff(fact_4409_log__inverse,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(log2(A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(log2(A),X)) ) ) ) ) ).

% log_inverse
tff(fact_4410_exp__divide__power__eq,axiom,
    ! [N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,N)))),N) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_4411_exp__divide__power__eq,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(power_power_real(aa_real_real(exp_real,aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_4412_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_4413_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_4414_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(divide_divide_real(aa_real_real(powr_real(X),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_4415_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_4416_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_4417_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(log2(B),X)) = aa_real_real(log2(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_4418_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log2(B),X)),Y) = aa_real_real(log2(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(powr_real(B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_4419_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(minus_minus_real(Y),aa_real_real(log2(B),X)) = aa_real_real(log2(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_4420_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_4421_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_4422_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_Code_integer(semiri3624122377584611663nteger,M) = one_one_Code_integer ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_Code_integer(semiri3624122377584611663nteger,M) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_4423_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_4424_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_4425_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_4426_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4427_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4428_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4429_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4430_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4431_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% pochhammer_same
tff(fact_4432_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% pochhammer_same
tff(fact_4433_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% pochhammer_same
tff(fact_4434_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% pochhammer_same
tff(fact_4435_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% pochhammer_same
tff(fact_4436_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_4437_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(minus_minus_real(aa_real_real(log2(B),X)),Y) = aa_real_real(log2(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_4438_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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)),N)) = 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)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_4439_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(B),X)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),aa_int_real(ring_1_of_int_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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_4440_tanh__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% tanh_altdef
tff(fact_4441_tanh__altdef,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% tanh_altdef
tff(fact_4442_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y5)),N)) = X )
             => pp(aa_real_bool(P,Y5)) ) ) ) ) ).

% split_root
tff(fact_4443_floor__divide__upper,axiom,
    ! [Q4: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P3),Q4)))),one_one_real)),Q4))) ) ).

% floor_divide_upper
tff(fact_4444_floor__divide__upper,axiom,
    ! [Q4: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P3),Q4)))),one_one_rat)),Q4))) ) ).

% floor_divide_upper
tff(fact_4445_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_4446_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_4447_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_4448_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_4449_sgn__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(sgn_sgn_complex,A)) = aa_complex_complex(sgn_sgn_complex,A) ).

% sgn_sgn
tff(fact_4450_sgn__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ).

% sgn_sgn
tff(fact_4451_sgn__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% sgn_sgn
tff(fact_4452_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_4453_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_4454_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_4455_mod__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% mod_0
tff(fact_4456_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_4457_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_4458_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_4459_mod__by__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,zero_z2226904508553997617atural) = A ).

% mod_by_0
tff(fact_4460_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_4461_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_4462_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_4463_mod__self,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,A) = zero_z2226904508553997617atural ).

% mod_self
tff(fact_4464_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_4465_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_4466_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_4467_bits__mod__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% bits_mod_0
tff(fact_4468_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_4469_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_4470_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_4471_mod__add__self2,axiom,
    ! [A: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),B) = modulo8411746178871703098atural(A,B) ).

% mod_add_self2
tff(fact_4472_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_4473_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_4474_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_4475_mod__add__self1,axiom,
    ! [B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_add_self1
tff(fact_4476_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_4477_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_4478_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_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(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_4479_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_4480_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_4481_sgn__0,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_4482_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_4483_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_4484_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_4485_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_4486_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_4487_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_4488_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_4489_sgn__1,axiom,
    aa_rat_rat(sgn_sgn_rat,one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_4490_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4491_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4492_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4493_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4494_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4495_inverse__sgn,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% inverse_sgn
tff(fact_4496_inverse__sgn,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% inverse_sgn
tff(fact_4497_sgn__inverse,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) ).

% sgn_inverse
tff(fact_4498_sgn__inverse,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% sgn_inverse
tff(fact_4499_sgn__inverse,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% sgn_inverse
tff(fact_4500_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) = X )
    <=> ? [N4: int] : X = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_floor_cancel
tff(fact_4501_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) = X )
    <=> ? [N4: int] : X = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_floor_cancel
tff(fact_4502_floor__of__int,axiom,
    ! [Z: int] : aa_real_int(archim6058952711729229775r_real,aa_int_real(ring_1_of_int_real,Z)) = Z ).

% floor_of_int
tff(fact_4503_floor__of__int,axiom,
    ! [Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_int_rat(ring_1_of_int_rat,Z)) = Z ).

% floor_of_int
tff(fact_4504_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_4505_tanh__minus,axiom,
    ! [X: real] : aa_real_real(tanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(tanh_real,X)) ).

% tanh_minus
tff(fact_4506_tanh__minus,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(tanh_complex,X)) ).

% tanh_minus
tff(fact_4507_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_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_4508_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_4509_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_4510_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] : modulo8411746178871703098atural(times_2397367101498566445atural(B,A),B) = zero_z2226904508553997617atural ).

% mod_mult_self1_is_0
tff(fact_4511_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_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_4512_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_4513_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_4514_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] : modulo8411746178871703098atural(times_2397367101498566445atural(A,B),B) = zero_z2226904508553997617atural ).

% mod_mult_self2_is_0
tff(fact_4515_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_4516_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_4517_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4518_mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% mod_by_1
tff(fact_4519_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_4520_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_4521_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_4522_bits__mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% bits_mod_by_1
tff(fact_4523_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_4524_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_4525_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_4526_mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] : divide5121882707175180666atural(modulo8411746178871703098atural(A,B),B) = zero_z2226904508553997617atural ).

% mod_div_trivial
tff(fact_4527_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4528_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_4529_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_4530_bits__mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] : divide5121882707175180666atural(modulo8411746178871703098atural(A,B),B) = zero_z2226904508553997617atural ).

% bits_mod_div_trivial
tff(fact_4531_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_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(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_4532_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_4533_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_4534_mod__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,C)),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self4
tff(fact_4535_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_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(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_4536_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_4537_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_4538_mod__mult__self3,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(C,B)),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self3
tff(fact_4539_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_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(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_4540_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_4541_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_4542_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(B,C)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self2
tff(fact_4543_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_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(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_4544_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_4545_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_4546_mod__mult__self1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(C,B)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self1
tff(fact_4547_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_4548_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_4549_sgn__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_less
tff(fact_4550_sgn__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_less
tff(fact_4551_sgn__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_less
tff(fact_4552_sgn__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_less
tff(fact_4553_sgn__greater,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_greater
tff(fact_4554_sgn__greater,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_greater
tff(fact_4555_sgn__greater,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_greater
tff(fact_4556_sgn__greater,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_greater
tff(fact_4557_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_4558_floor__zero,axiom,
    aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_4559_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_4560_floor__one,axiom,
    aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).

% floor_one
tff(fact_4561_floor__of__nat,axiom,
    ! [N: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% floor_of_nat
tff(fact_4562_floor__of__nat,axiom,
    ! [N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% floor_of_nat
tff(fact_4563_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4564_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4565_sgn__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_4566_sgn__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_4567_sgn__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_4568_sgn__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_4569_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_4570_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_4571_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_4572_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_4573_floor__uminus__of__int,axiom,
    ! [Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(uminus_uminus_int,Z) ).

% floor_uminus_of_int
tff(fact_4574_floor__uminus__of__int,axiom,
    ! [Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(uminus_uminus_int,Z) ).

% floor_uminus_of_int
tff(fact_4575_floor__diff__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(minus_minus_real(X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) ).

% floor_diff_of_int
tff(fact_4576_floor__diff__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) ).

% floor_diff_of_int
tff(fact_4577_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_4578_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_4579_floor__add2,axiom,
    ! [X: real,Y: real] :
      ( ( member_real(X,ring_1_Ints_real)
        | member_real(Y,ring_1_Ints_real) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) ) ).

% floor_add2
tff(fact_4580_floor__add2,axiom,
    ! [X: rat,Y: rat] :
      ( ( member_rat(X,ring_1_Ints_rat)
        | member_rat(Y,ring_1_Ints_rat) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ) ).

% floor_add2
tff(fact_4581_sgn__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_4582_sgn__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_4583_sgn__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_4584_sgn__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_4585_zero__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_floor
tff(fact_4586_zero__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X)) ) ).

% zero_le_floor
tff(fact_4587_floor__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% floor_less_zero
tff(fact_4588_floor__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat)) ) ).

% floor_less_zero
tff(fact_4589_zero__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% zero_less_floor
tff(fact_4590_zero__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% zero_less_floor
tff(fact_4591_floor__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_le_zero
tff(fact_4592_floor__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_le_zero
tff(fact_4593_one__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% one_le_floor
tff(fact_4594_one__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% one_le_floor
tff(fact_4595_floor__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_less_one
tff(fact_4596_floor__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_less_one
tff(fact_4597_floor__diff__one,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(minus_minus_real(X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) ).

% floor_diff_one
tff(fact_4598_floor__diff__one,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ).

% floor_diff_one
tff(fact_4599_of__nat__mod,axiom,
    ! [M: nat,N: nat] : semiri3763490453095760265atural(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = modulo8411746178871703098atural(semiri3763490453095760265atural(M),semiri3763490453095760265atural(N)) ).

% of_nat_mod
tff(fact_4600_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mod
tff(fact_4601_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mod
tff(fact_4602_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mod
tff(fact_4603_tanh__real__bounds,axiom,
    ! [X: real] : member_real(aa_real_real(tanh_real,X),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_4604_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_4605_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_4606_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_4607_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_4608_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_4609_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_4610_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_4611_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_4612_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_4613_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_4614_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_4615_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),C) ).

% mod_add_eq
tff(fact_4616_mod__add__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C)),modulo8411746178871703098atural(B,C)),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) ).

% mod_add_eq
tff(fact_4617_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A5),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B5),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_4618_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A5),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_4619_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A5,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A5),B5),C) ) ) ) ).

% mod_add_cong
tff(fact_4620_mod__add__cong,axiom,
    ! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B5: code_natural] :
      ( ( modulo8411746178871703098atural(A,C) = modulo8411746178871703098atural(A5,C) )
     => ( ( modulo8411746178871703098atural(B,C) = modulo8411746178871703098atural(B5,C) )
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A5),B5),C) ) ) ) ).

% mod_add_cong
tff(fact_4621_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_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_4622_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_4623_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),C) ).

% mod_add_left_eq
tff(fact_4624_mod__add__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C)),B),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) ).

% mod_add_left_eq
tff(fact_4625_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_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_4626_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_4627_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),C) ).

% mod_add_right_eq
tff(fact_4628_mod__add__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),modulo8411746178871703098atural(B,C)),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) ).

% mod_add_right_eq
tff(fact_4629_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_4630_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),C) ).

% mod_diff_eq
tff(fact_4631_mod__diff__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A5),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B5),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A5),B5)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4632_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A5,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A5),B5),C) ) ) ) ).

% mod_diff_cong
tff(fact_4633_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_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_4634_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),C) ).

% mod_diff_left_eq
tff(fact_4635_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_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_4636_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),C) ).

% mod_diff_right_eq
tff(fact_4637_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_4638_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).

% sgn_mult
tff(fact_4639_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_4640_sgn__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ).

% sgn_mult
tff(fact_4641_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_4642_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_4643_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_4644_mod__minus__cong,axiom,
    ! [A: int,B: int,A5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A5),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A5)),B) ) ) ).

% mod_minus_cong
tff(fact_4645_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A5: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A5,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A5),B) ) ) ).

% mod_minus_cong
tff(fact_4646_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_4647_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_4648_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ) ) ).

% same_sgn_sgn_add
tff(fact_4649_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_4650_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(sgn_sgn_rat,A) ) ) ).

% same_sgn_sgn_add
tff(fact_4651_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_4652_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))) ) ).

% floor_mono
tff(fact_4653_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))) ) ).

% floor_mono
tff(fact_4654_of__int__floor__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X)) ).

% of_int_floor_le
tff(fact_4655_of__int__floor__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X)) ).

% of_int_floor_le
tff(fact_4656_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% floor_less_cancel
tff(fact_4657_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% floor_less_cancel
tff(fact_4658_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4659_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4660_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4661_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4662_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4663_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4664_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4665_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_4666_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_4667_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = A )
    <=> ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4668_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D4: 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),D4)) ) ).

% mod_eqE
tff(fact_4669_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D4: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D4)) ) ).

% mod_eqE
tff(fact_4670_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_4671_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_4672_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,divide6298287555418463151nteger(A,C)),divide6298287555418463151nteger(B,C))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_4673_div__add1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C)),divide5121882707175180666atural(B,C))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C)),modulo8411746178871703098atural(B,C)),C)) ).

% div_add1_eq
tff(fact_4674_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4675_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4676_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4677_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4678_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_4679_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_4680_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_4681_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_4682_sgn__minus__1,axiom,
    aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_4683_mult__sgn__abs,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)),abs_abs_Code_integer(X)) = X ).

% mult_sgn_abs
tff(fact_4684_mult__sgn__abs,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),aa_real_real(abs_abs_real,X)) = X ).

% mult_sgn_abs
tff(fact_4685_mult__sgn__abs,axiom,
    ! [X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,X)),aa_rat_rat(abs_abs_rat,X)) = X ).

% mult_sgn_abs
tff(fact_4686_mult__sgn__abs,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ).

% mult_sgn_abs
tff(fact_4687_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_4688_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_4689_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_4690_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(abs_abs_rat,A)) = A ).

% sgn_mult_abs
tff(fact_4691_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_4692_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_4693_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_4694_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_4695_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = A ).

% abs_mult_sgn
tff(fact_4696_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_4697_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4698_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4699_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : aa_rat_rat(abs_abs_rat,K) = aa_rat_rat(times_times_rat(K),aa_rat_rat(sgn_sgn_rat,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4700_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4701_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% same_sgn_abs_add
tff(fact_4702_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( 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)) ) ) ).

% same_sgn_abs_add
tff(fact_4703_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( 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)) ) ) ).

% same_sgn_abs_add
tff(fact_4704_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( 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)) ) ) ).

% same_sgn_abs_add
tff(fact_4705_floor__le__ceiling,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim7802044766580827645g_real,X))) ).

% floor_le_ceiling
tff(fact_4706_floor__le__ceiling,axiom,
    ! [X: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X))) ).

% floor_le_ceiling
tff(fact_4707_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_4708_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L))) ) ).

% neg_mod_bound
tff(fact_4709_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_4710_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_4711_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_4712_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
    <=> ? [Q6: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q6) ) ).

% zmod_eq_0_iff
tff(fact_4713_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
     => ? [Q2: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q2) ) ).

% zmod_eq_0D
tff(fact_4714_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_4715_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% le_floor_iff
tff(fact_4716_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% le_floor_iff
tff(fact_4717_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% floor_less_iff
tff(fact_4718_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% floor_less_iff
tff(fact_4719_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4720_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4721_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4722_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4723_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4724_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4725_le__floor__add,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))) ).

% le_floor_add
tff(fact_4726_le__floor__add,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)))) ).

% le_floor_add
tff(fact_4727_int__add__floor,axiom,
    ! [Z: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).

% int_add_floor
tff(fact_4728_int__add__floor,axiom,
    ! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ).

% int_add_floor
tff(fact_4729_floor__add__int,axiom,
    ! [X: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_4730_floor__add__int,axiom,
    ! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_4731_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_1_pos
tff(fact_4732_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_1_pos
tff(fact_4733_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_1_pos
tff(fact_4734_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_1_pos
tff(fact_4735_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_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(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_4736_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_4737_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_4738_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : divide5121882707175180666atural(times_2397367101498566445atural(A,B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(A,divide5121882707175180666atural(B,C))),divide5121882707175180666atural(times_2397367101498566445atural(A,modulo8411746178871703098atural(B,C)),C)) ).

% div_mult1_eq
tff(fact_4739_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(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_4740_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_4741_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4742_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4743_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))) = A ).

% mod_mult_div_eq
tff(fact_4744_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_4745_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4746_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4747_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4748_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_4749_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4750_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)) = A ).

% mod_div_mult_eq
tff(fact_4751_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(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_4752_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_4753_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4754_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)),modulo8411746178871703098atural(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4755_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(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_4756_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_4757_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_4758_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] : A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)),modulo8411746178871703098atural(A,B)) ).

% mod_div_decomp
tff(fact_4759_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(divide_divide_int(A),B)),B)),aa_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_4760_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_4761_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)),modulo364778990260209775nteger(A,B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4762_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)),modulo8411746178871703098atural(A,B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4763_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(divide_divide_int(A),B))),aa_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_4764_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_4765_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))),modulo364778990260209775nteger(A,B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4766_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4767_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(divide_divide_int(A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_4768_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_4769_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4770_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))) = modulo8411746178871703098atural(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4771_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(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4772_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_4773_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4774_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),modulo8411746178871703098atural(A,B)) = times_2397367101498566445atural(B,divide5121882707175180666atural(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4775_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(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4776_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_4777_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4778_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),modulo8411746178871703098atural(A,B)) = times_2397367101498566445atural(divide5121882707175180666atural(A,B),B) ).

% minus_mod_eq_div_mult
tff(fact_4779_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(divide_divide_int(A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_4780_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_4781_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4782_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)) = modulo8411746178871703098atural(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4783_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,K)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(divide_divide_int(K),L) ).

% floor_divide_of_int_eq
tff(fact_4784_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(divide_divide_int(K),L) ).

% floor_divide_of_int_eq
tff(fact_4785_ceiling__def,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(uminus_uminus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,X))) ).

% ceiling_def
tff(fact_4786_ceiling__def,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X))) ).

% ceiling_def
tff(fact_4787_floor__minus,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,X)) = aa_int_int(uminus_uminus_int,aa_real_int(archim7802044766580827645g_real,X)) ).

% floor_minus
tff(fact_4788_floor__minus,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim2889992004027027881ng_rat,X)) ).

% floor_minus
tff(fact_4789_ceiling__minus,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,X)) = aa_int_int(uminus_uminus_int,aa_real_int(archim6058952711729229775r_real,X)) ).

% ceiling_minus
tff(fact_4790_ceiling__minus,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,X)) ).

% ceiling_minus
tff(fact_4791_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_4792_floor__power,axiom,
    ! [X: real,N: nat] :
      ( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(power_power_real(X),N)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,X)),N) ) ) ).

% floor_power
tff(fact_4793_floor__power,axiom,
    ! [X: rat,N: nat] :
      ( ( X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(X),N)) = aa_nat_int(power_power_int(aa_rat_int(archim3151403230148437115or_rat,X)),N) ) ) ).

% floor_power
tff(fact_4794_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
tff(fact_4795_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ) ).

% abs_sgn_eq
tff(fact_4796_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ) ).

% abs_sgn_eq
tff(fact_4797_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ) ).

% abs_sgn_eq
tff(fact_4798_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_4799_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,N),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_4800_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo8411746178871703098atural(semiri2447717529341329178atural(N),semiri2447717529341329178atural(M)) = zero_z2226904508553997617atural ) ) ).

% fact_mod
tff(fact_4801_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_4802_frac__def,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,X) = aa_real_real(minus_minus_real(X),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))) ).

% frac_def
tff(fact_4803_frac__def,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,X) = aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))) ).

% frac_def
tff(fact_4804_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_4805_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_4806_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B))) ) ) ).

% neg_mod_conj
tff(fact_4807_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_4808_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_4809_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(modulo_modulo_int(A2),N) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A2),N)),aa_int_int(divide_divide_int(B2),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_4810_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_4811_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_4812_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_4813_div__mod__decomp__int,axiom,
    ! [A2: int,N: 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(divide_divide_int(A2),N)),N)),aa_int_int(modulo_modulo_int(A2),N)) ).

% div_mod_decomp_int
tff(fact_4814_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X))) ).

% tanh_real_gt_neg1
tff(fact_4815_of__nat__floor,axiom,
    ! [R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2)) ) ).

% of_nat_floor
tff(fact_4816_of__nat__floor,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R2)))),R2)) ) ).

% of_nat_floor
tff(fact_4817_one__add__floor,axiom,
    ! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% one_add_floor
tff(fact_4818_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) ).

% one_add_floor
tff(fact_4819_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ).

% le_mult_nat_floor
tff(fact_4820_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,A))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))))) ).

% le_mult_nat_floor
tff(fact_4821_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% floor_divide_of_nat_eq
tff(fact_4822_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% floor_divide_of_nat_eq
tff(fact_4823_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_4824_sgn__real__def,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_4825_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_1_neg
tff(fact_4826_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_1_neg
tff(fact_4827_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_1_neg
tff(fact_4828_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_1_neg
tff(fact_4829_sgn__if,axiom,
    ! [X: int] :
      ( ( ( X = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,X) = zero_zero_int ) )
      & ( ( X != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
           => ( aa_int_int(sgn_sgn_int,X) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% sgn_if
tff(fact_4830_sgn__if,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,X) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
           => ( aa_real_real(sgn_sgn_real,X) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_if
tff(fact_4831_sgn__if,axiom,
    ! [X: code_integer] :
      ( ( ( X = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = zero_z3403309356797280102nteger ) )
      & ( ( X != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = one_one_Code_integer ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ) ) ).

% sgn_if
tff(fact_4832_sgn__if,axiom,
    ! [X: rat] :
      ( ( ( X = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,X) = zero_zero_rat ) )
      & ( ( X != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
           => ( aa_rat_rat(sgn_sgn_rat,X) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
           => ( aa_rat_rat(sgn_sgn_rat,X) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_if
tff(fact_4833_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N: nat] : modulo8411746178871703098atural(A,times_2397367101498566445atural(semiri3763490453095760265atural(M),semiri3763490453095760265atural(N))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(semiri3763490453095760265atural(M),modulo8411746178871703098atural(divide5121882707175180666atural(A,semiri3763490453095760265atural(M)),semiri3763490453095760265atural(N)))),modulo8411746178871703098atural(A,semiri3763490453095760265atural(M))) ).

% mod_mult2_eq'
tff(fact_4834_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_4835_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_4836_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_4837_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_4838_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_4839_ceiling__altdef,axiom,
    ! [X: real] :
      ( ( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) )
      & ( ( X != aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) ) ) ) ).

% ceiling_altdef
tff(fact_4840_ceiling__altdef,axiom,
    ! [X: rat] :
      ( ( ( X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) )
      & ( ( X != aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ) ) ) ).

% ceiling_altdef
tff(fact_4841_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim6058952711729229775r_real,X))),one_one_int)) ).

% ceiling_diff_floor_le_1
tff(fact_4842_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X))),one_one_int)) ).

% ceiling_diff_floor_le_1
tff(fact_4843_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq
tff(fact_4844_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_4845_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_4846_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_4847_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_4848_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_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_4849_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4850_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,X)),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(divide_divide_int(X),D))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(X),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_4851_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Z ) ) ) ).

% floor_unique
tff(fact_4852_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).

% floor_unique
tff(fact_4853_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real))) ) ) ).

% floor_eq_iff
tff(fact_4854_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat))) ) ) ).

% floor_eq_iff
tff(fact_4855_floor__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim6058952711729229775r_real,T)))
    <=> ! [I3: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I3)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I3)),one_one_real))) )
         => pp(aa_int_bool(P,I3)) ) ) ).

% floor_split
tff(fact_4856_floor__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim3151403230148437115or_rat,T)))
    <=> ! [I3: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I3)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat))) )
         => pp(aa_int_bool(P,I3)) ) ) ).

% floor_split
tff(fact_4857_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor
tff(fact_4858_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor
tff(fact_4859_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).

% less_floor_iff
tff(fact_4860_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% less_floor_iff
tff(fact_4861_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% floor_le_iff
tff(fact_4862_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% floor_le_iff
tff(fact_4863_floor__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_4864_floor__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_4865_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C))),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4866_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4867_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_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(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4868_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_4869_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_4870_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq2
tff(fact_4871_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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(divide_divide_int(aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_4872_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4873_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4874_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(modulo_modulo_int(N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_4875_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_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(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_4876_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_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_4877_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_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(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_4878_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_4879_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_4880_floor__divide__lower,axiom,
    ! [Q4: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q4))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P3),Q4)))),Q4)),P3)) ) ).

% floor_divide_lower
tff(fact_4881_floor__divide__lower,axiom,
    ! [Q4: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q4))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P3),Q4)))),Q4)),P3)) ) ).

% floor_divide_lower
tff(fact_4882_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4883_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4884_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_rat(ring_1_of_int_rat,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4885_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( member_real(A,ring_1_Ints_real)
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_int(ring_1_of_int_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4886_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_real(ring_1_of_int_real,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4887_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4888_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4889_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( member_rat(A,ring_1_Ints_rat)
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_int(ring_1_of_int_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))))) ) ) ).

% le_mult_floor_Ints
tff(fact_4890_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_4891_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_4892_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_4893_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_4894_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A2),N)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B2),N)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(divide_divide_int(B2),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_4895_sgn__one,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_one
tff(fact_4896_sgn__one,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_one
tff(fact_4897_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_4898_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_4899_Real__Vector__Spaces_Osgn__minus,axiom,
    ! [X: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,X)) ).

% Real_Vector_Spaces.sgn_minus
tff(fact_4900_Real__Vector__Spaces_Osgn__minus,axiom,
    ! [X: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sgn_sgn_complex,X)) ).

% Real_Vector_Spaces.sgn_minus
tff(fact_4901_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X) = zero_zero_complex )
    <=> ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_4902_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sgn_sgn_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_4903_tanh__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cosh_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y)))) ) ) ) ).

% tanh_add
tff(fact_4904_tanh__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cosh_real,X) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))) ) ) ) ).

% tanh_add
tff(fact_4905_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ).

% verit_le_mono_div
tff(fact_4906_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X7,M4)),aa_nat_real(X7,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_4907_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_4908_cosh__minus,axiom,
    ! [X: real] : aa_real_real(cosh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(cosh_real,X) ).

% cosh_minus
tff(fact_4909_cosh__minus,axiom,
    ! [X: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(cosh_complex,X) ).

% cosh_minus
tff(fact_4910_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_4911_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_4912_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_4913_Suc__mod__mult__self4,axiom,
    ! [N: 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,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_4914_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: 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),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_4915_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: 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,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_4916_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: 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),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_4917_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: 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),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_4918_mod__Suc__eq,axiom,
    ! [M: nat,N: 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),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_4919_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L2: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_4920_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_4921_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_4922_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M2),zero_zero_nat))
     => ( ! [M2: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M2),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_4923_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_4924_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_4925_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q2: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_4926_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_4927_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_4928_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_4929_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X))) ).

% cosh_real_ge_1
tff(fact_4930_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_4931_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(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_4932_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_4933_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ) ) ).

% div_less_mono
tff(fact_4934_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_4935_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S2: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_4936_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q2: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4937_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q4))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_4938_div__mod__decomp,axiom,
    ! [A2: nat,N: 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),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).

% div_mod_decomp
tff(fact_4939_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = 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),N)),N)) ).

% modulo_nat_def
tff(fact_4940_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_4941_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_4942_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% field_char_0_class.of_nat_div
tff(fact_4943_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = 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),N)))),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% field_char_0_class.of_nat_div
tff(fact_4944_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_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),N)))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% field_char_0_class.of_nat_div
tff(fact_4945_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_4946_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,X)),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,X),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_4947_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(modulo_modulo_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_4948_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_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_4949_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4950_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_4951_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_4952_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_4953_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_4954_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Z),M)),aa_nat_real(power_power_real(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z),W))))) ) ) ).

% norm_power_diff
tff(fact_4955_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W))))) ) ) ).

% norm_power_diff
tff(fact_4956_count__notin,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ~ member_complex(X,set_complex2(Xs))
     => ( aa_complex_nat(count_list_complex(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4957_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ member_real(X,set_real2(Xs))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4958_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ member_set_nat(X,set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4959_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => ( count_list_VEBT_VEBT(Xs,X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4960_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( aa_nat_nat(count_list_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4961_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ member_int(X,set_int2(Xs))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4962_metric__Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V975177566351809787t_real(aa_nat_real(X7,M4)),aa_nat_real(X7,N4))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_4963_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( ( X != zero_zero_real )
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N)) )
       => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X),N) ) ) ) ).

% powr_real_of_int'
tff(fact_4964_round__diff__minimal,axiom,
    ! [Z: real,M: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_4965_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,M))))) ).

% round_diff_minimal
tff(fact_4966_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_4967_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_4968_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,M) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_4969_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_4970_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_4971_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_4972_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_4973_power__int__mult__numeral,axiom,
    ! [X: real,M: num,N: num] : aa_int_real(power_int_real(aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).

% power_int_mult_numeral
tff(fact_4974_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_4975_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_4976_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_4977_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_4978_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_4979_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_4980_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_4981_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_4982_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_4983_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_4984_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_4985_numeral__times__numeral,axiom,
    ! [M: num,N: 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,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_4986_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_4987_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_4988_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_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_4989_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_4990_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_4991_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_4992_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_4993_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_4994_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_4995_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_4996_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_4997_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_4998_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_4999_numeral__plus__numeral,axiom,
    ! [M: num,N: 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,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_5000_numeral__plus__numeral,axiom,
    ! [M: num,N: 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,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_5001_numeral__plus__numeral,axiom,
    ! [M: num,N: 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,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_5002_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_5003_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_5004_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_5005_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_5006_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_5007_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_5008_neg__numeral__eq__iff,axiom,
    ! [M: num,N: 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,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_5009_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_5010_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_5011_neg__numeral__eq__iff,axiom,
    ! [M: num,N: 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,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_5012_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).

% of_nat_numeral
tff(fact_5013_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_5014_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = aa_num_real(numeral_numeral_real,N) ).

% of_nat_numeral
tff(fact_5015_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_5016_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,N)) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% of_nat_numeral
tff(fact_5017_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% of_nat_numeral
tff(fact_5018_abs__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% abs_numeral
tff(fact_5019_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,N) ).

% abs_numeral
tff(fact_5020_abs__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_numeral
tff(fact_5021_abs__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).

% abs_numeral
tff(fact_5022_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = aa_num_Code_integer(numera6620942414471956472nteger,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5023_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5024_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5025_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5026_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5027_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_5028_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_5029_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_5030_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_5031_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_5032_norm__minus__cancel,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(real_V7735802525324610683m_real,X) ).

% norm_minus_cancel
tff(fact_5033_norm__minus__cancel,axiom,
    ! [X: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_real(real_V1022390504157884413omplex,X) ).

% norm_minus_cancel
tff(fact_5034_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_5035_floor__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_5036_power__int__add__numeral,axiom,
    ! [X: real,M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,M))),aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,N))) = aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N))) ).

% power_int_add_numeral
tff(fact_5037_power__int__add__numeral,axiom,
    ! [X: rat,M: num,N: num] : aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,M))),aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,N))) = aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N))) ).

% power_int_add_numeral
tff(fact_5038_power__int__add__numeral2,axiom,
    ! [X: real,M: num,N: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,N))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)))),B) ).

% power_int_add_numeral2
tff(fact_5039_power__int__add__numeral2,axiom,
    ! [X: rat,M: num,N: num,B: rat] : aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,M))),aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,N))),B)) = aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)))),B) ).

% power_int_add_numeral2
tff(fact_5040_power__int__1__left,axiom,
    ! [N: int] : aa_int_complex(power_int_complex(one_one_complex),N) = one_one_complex ).

% power_int_1_left
tff(fact_5041_power__int__1__left,axiom,
    ! [N: int] : aa_int_real(power_int_real(one_one_real),N) = one_one_real ).

% power_int_1_left
tff(fact_5042_power__int__1__left,axiom,
    ! [N: int] : aa_int_rat(power_int_rat(one_one_rat),N) = one_one_rat ).

% power_int_1_left
tff(fact_5043_dist__add__cancel,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(real_V975177566351809787t_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(real_V975177566351809787t_real(B),C) ).

% dist_add_cancel
tff(fact_5044_dist__add__cancel2,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(real_V975177566351809787t_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)) = aa_real_real(real_V975177566351809787t_real(B),C) ).

% dist_add_cancel2
tff(fact_5045_power__int__numeral,axiom,
    ! [X: real,N: num] : aa_int_real(power_int_real(X),aa_num_int(numeral_numeral_int,N)) = aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,N)) ).

% power_int_numeral
tff(fact_5046_power__int__numeral,axiom,
    ! [X: complex,N: num] : aa_int_complex(power_int_complex(X),aa_num_int(numeral_numeral_int,N)) = aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,N)) ).

% power_int_numeral
tff(fact_5047_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_5048_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_5049_power__int__1__right,axiom,
    ! [Y: real] : aa_int_real(power_int_real(Y),one_one_int) = Y ).

% power_int_1_right
tff(fact_5050_power__int__sgn,axiom,
    ! [A: real,N: int] : aa_real_real(sgn_sgn_real,aa_int_real(power_int_real(A),N)) = aa_int_real(power_int_real(aa_real_real(sgn_sgn_real,A)),N) ).

% power_int_sgn
tff(fact_5051_power__int__sgn,axiom,
    ! [A: rat,N: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(power_int_rat(A),N)) = aa_int_rat(power_int_rat(aa_rat_rat(sgn_sgn_rat,A)),N) ).

% power_int_sgn
tff(fact_5052_round__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5053_round__numeral,axiom,
    ! [N: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5054_round__of__int,axiom,
    ! [N: int] : aa_real_int(archim8280529875227126926d_real,aa_int_real(ring_1_of_int_real,N)) = N ).

% round_of_int
tff(fact_5055_round__of__int,axiom,
    ! [N: int] : aa_rat_int(archim7778729529865785530nd_rat,aa_int_rat(ring_1_of_int_rat,N)) = N ).

% round_of_int
tff(fact_5056_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5057_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5058_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5059_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_5060_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(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_5061_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_5062_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_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(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_5063_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_5064_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_5065_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(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_5066_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_5067_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_5068_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_5069_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_5070_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5071_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5072_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5073_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_5074_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(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_5075_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_5076_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_5077_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_5078_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(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_5079_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_5080_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_5081_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_5082_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_5083_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_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,N))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_5084_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_5085_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_5086_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_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,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_5087_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_5088_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_5089_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_5090_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_5091_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_5092_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_5093_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_5094_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_5095_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_5096_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_5097_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_5098_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_5099_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_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_5100_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_5101_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_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_5102_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_5103_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_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_5104_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(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5105_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5106_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_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_5107_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_5108_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Y)) = aa_real_real(aa_rea7340605660300375871l_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_5109_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5110_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5111_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_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_5112_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_5113_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: 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,N))) = 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,N))) ).

% add_neg_numeral_simps(3)
tff(fact_5114_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N))) ).

% add_neg_numeral_simps(3)
tff(fact_5115_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% add_neg_numeral_simps(3)
tff(fact_5116_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = 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,N))) ).

% add_neg_numeral_simps(3)
tff(fact_5117_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_5118_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_5119_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5120_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5121_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_5122_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_5123_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: 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,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_5124_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_5125_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_5126_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_5127_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_5128_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: 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,N))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_5129_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_5130_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_5131_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_5132_abs__neg__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).

% abs_neg_numeral
tff(fact_5133_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,N) ).

% abs_neg_numeral
tff(fact_5134_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% abs_neg_numeral
tff(fact_5135_abs__neg__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_neg_numeral
tff(fact_5136_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_5137_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_5138_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_5139_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_5140_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_5141_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_5142_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_5143_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_5144_power__int__mult__distrib__numeral2,axiom,
    ! [X: complex,W: num,M: int] : aa_int_complex(power_int_complex(aa_complex_complex(times_times_complex(X),numera6690914467698888265omplex(W))),M) = aa_complex_complex(times_times_complex(aa_int_complex(power_int_complex(X),M)),aa_int_complex(power_int_complex(numera6690914467698888265omplex(W)),M)) ).

% power_int_mult_distrib_numeral2
tff(fact_5145_power__int__mult__distrib__numeral2,axiom,
    ! [X: real,W: num,M: int] : aa_int_real(power_int_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_num_real(numeral_numeral_real,W))),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(aa_num_real(numeral_numeral_real,W)),M)) ).

% power_int_mult_distrib_numeral2
tff(fact_5146_power__int__mult__distrib__numeral2,axiom,
    ! [X: rat,W: num,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(times_times_rat(X),aa_num_rat(numeral_numeral_rat,W))),M) = aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(aa_num_rat(numeral_numeral_rat,W)),M)) ).

% power_int_mult_distrib_numeral2
tff(fact_5147_power__int__mult__distrib__numeral1,axiom,
    ! [W: num,Y: complex,M: int] : aa_int_complex(power_int_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)),M) = aa_complex_complex(times_times_complex(aa_int_complex(power_int_complex(numera6690914467698888265omplex(W)),M)),aa_int_complex(power_int_complex(Y),M)) ).

% power_int_mult_distrib_numeral1
tff(fact_5148_power__int__mult__distrib__numeral1,axiom,
    ! [W: num,Y: real,M: int] : aa_int_real(power_int_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Y)),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(aa_num_real(numeral_numeral_real,W)),M)),aa_int_real(power_int_real(Y),M)) ).

% power_int_mult_distrib_numeral1
tff(fact_5149_power__int__mult__distrib__numeral1,axiom,
    ! [W: num,Y: rat,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y)),M) = aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(aa_num_rat(numeral_numeral_rat,W)),M)),aa_int_rat(power_int_rat(Y),M)) ).

% power_int_mult_distrib_numeral1
tff(fact_5150_power__int__0__left,axiom,
    ! [M: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_real(power_int_real(zero_zero_real),M) = zero_zero_real ) ) ).

% power_int_0_left
tff(fact_5151_power__int__0__left,axiom,
    ! [M: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_rat(power_int_rat(zero_zero_rat),M) = zero_zero_rat ) ) ).

% power_int_0_left
tff(fact_5152_power__int__eq__0__iff,axiom,
    ! [X: real,N: int] :
      ( ( aa_int_real(power_int_real(X),N) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( N != zero_zero_int ) ) ) ).

% power_int_eq_0_iff
tff(fact_5153_power__int__eq__0__iff,axiom,
    ! [X: rat,N: int] :
      ( ( aa_int_rat(power_int_rat(X),N) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( N != zero_zero_int ) ) ) ).

% power_int_eq_0_iff
tff(fact_5154_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_5155_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_num_real(numeral_numeral_real,W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_5156_power__int__0__right,axiom,
    ! [X: complex] : aa_int_complex(power_int_complex(X),zero_zero_int) = one_one_complex ).

% power_int_0_right
tff(fact_5157_power__int__0__right,axiom,
    ! [X: real] : aa_int_real(power_int_real(X),zero_zero_int) = one_one_real ).

% power_int_0_right
tff(fact_5158_power__int__0__right,axiom,
    ! [X: rat] : aa_int_rat(power_int_rat(X),zero_zero_int) = one_one_rat ).

% power_int_0_right
tff(fact_5159_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_5160_dist__0__norm,axiom,
    ! [X: real] : aa_real_real(real_V975177566351809787t_real(zero_zero_real),X) = aa_real_real(real_V7735802525324610683m_real,X) ).

% dist_0_norm
tff(fact_5161_dist__0__norm,axiom,
    ! [X: complex] : aa_complex_real(real_V3694042436643373181omplex(zero_zero_complex),X) = aa_complex_real(real_V1022390504157884413omplex,X) ).

% dist_0_norm
tff(fact_5162_abs__power__int__minus,axiom,
    ! [A: real,N: int] : aa_real_real(abs_abs_real,aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,A)),N)) = aa_real_real(abs_abs_real,aa_int_real(power_int_real(A),N)) ).

% abs_power_int_minus
tff(fact_5163_abs__power__int__minus,axiom,
    ! [A: rat,N: int] : aa_rat_rat(abs_abs_rat,aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_rat_rat(abs_abs_rat,aa_int_rat(power_int_rat(A),N)) ).

% abs_power_int_minus
tff(fact_5164_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_5165_norm__of__nat,axiom,
    ! [N: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% norm_of_nat
tff(fact_5166_norm__of__nat,axiom,
    ! [N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(semiri8010041392384452111omplex,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% norm_of_nat
tff(fact_5167_dist__diff_I2_J,axiom,
    ! [A: real,B: real] : aa_real_real(real_V975177566351809787t_real(aa_real_real(minus_minus_real(A),B)),A) = aa_real_real(real_V7735802525324610683m_real,B) ).

% dist_diff(2)
tff(fact_5168_dist__diff_I2_J,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V3694042436643373181omplex(aa_complex_complex(minus_minus_complex(A),B)),A) = aa_complex_real(real_V1022390504157884413omplex,B) ).

% dist_diff(2)
tff(fact_5169_dist__diff_I1_J,axiom,
    ! [A: real,B: real] : aa_real_real(real_V975177566351809787t_real(A),aa_real_real(minus_minus_real(A),B)) = aa_real_real(real_V7735802525324610683m_real,B) ).

% dist_diff(1)
tff(fact_5170_dist__diff_I1_J,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V3694042436643373181omplex(A),aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_real(real_V1022390504157884413omplex,B) ).

% dist_diff(1)
tff(fact_5171_power__int__of__nat,axiom,
    ! [X: real,N: nat] : aa_int_real(power_int_real(X),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_real(power_power_real(X),N) ).

% power_int_of_nat
tff(fact_5172_power__int__of__nat,axiom,
    ! [X: complex,N: nat] : aa_int_complex(power_int_complex(X),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_complex(power_power_complex(X),N) ).

% power_int_of_nat
tff(fact_5173_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_5174_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_5175_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_5176_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5177_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5178_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5179_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_5180_round__of__nat,axiom,
    ! [N: nat] : aa_real_int(archim8280529875227126926d_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% round_of_nat
tff(fact_5181_round__of__nat,axiom,
    ! [N: nat] : aa_rat_int(archim7778729529865785530nd_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% round_of_nat
tff(fact_5182_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(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_5183_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_5184_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_5185_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_5186_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_5187_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_5188_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)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) = B ) )
        & ( ( aa_num_real(numeral_numeral_real,W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_5189_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_5190_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_5191_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 )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_5192_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_5193_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_5194_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_5195_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_5196_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_5197_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,W)) ).

% inverse_eq_divide_numeral
tff(fact_5198_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).

% inverse_eq_divide_numeral
tff(fact_5199_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,W)) ).

% inverse_eq_divide_numeral
tff(fact_5200_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_5201_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_5202_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5203_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5204_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5205_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5206_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5207_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5208_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5209_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5210_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5211_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5212_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5213_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5214_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5215_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5216_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5217_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5218_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5219_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5220_power__int__minus__one__mult__self_H,axiom,
    ! [M: int,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),M)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),M)),B)) = B ).

% power_int_minus_one_mult_self'
tff(fact_5221_power__int__minus__one__mult__self_H,axiom,
    ! [M: int,B: complex] : aa_complex_complex(times_times_complex(aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)),aa_complex_complex(times_times_complex(aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)),B)) = B ).

% power_int_minus_one_mult_self'
tff(fact_5222_power__int__minus__one__mult__self_H,axiom,
    ! [M: int,B: rat] : aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),B)) = B ).

% power_int_minus_one_mult_self'
tff(fact_5223_power__int__minus__one__mult__self,axiom,
    ! [M: int] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),M)),aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),M)) = one_one_real ).

% power_int_minus_one_mult_self
tff(fact_5224_power__int__minus__one__mult__self,axiom,
    ! [M: int] : aa_complex_complex(times_times_complex(aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)),aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)) = one_one_complex ).

% power_int_minus_one_mult_self
tff(fact_5225_power__int__minus__one__mult__self,axiom,
    ! [M: int] : aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)) = one_one_rat ).

% power_int_minus_one_mult_self
tff(fact_5226_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5227_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5228_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5229_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5230_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5231_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5232_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5233_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5234_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5235_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5236_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5237_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5238_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_le_floor
tff(fact_5239_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_le_floor
tff(fact_5240_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_numeral
tff(fact_5241_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_numeral
tff(fact_5242_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_numeral
tff(fact_5243_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_numeral
tff(fact_5244_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5245_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5246_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_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)) ).

% floor_neg_numeral
tff(fact_5247_floor__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_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)) ).

% floor_neg_numeral
tff(fact_5248_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5249_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5250_floor__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(minus_minus_real(X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_5251_floor__diff__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_5252_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_5253_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_5254_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5255_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5256_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5257_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5258_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5259_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5260_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5261_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5262_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5263_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5264_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5265_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5266_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: 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)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_5267_floor__numeral__power,axiom,
    ! [X: num,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% floor_numeral_power
tff(fact_5268_floor__numeral__power,axiom,
    ! [X: num,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% floor_numeral_power
tff(fact_5269_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% ceiling_numeral_power
tff(fact_5270_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% ceiling_numeral_power
tff(fact_5271_power__int__minus1__right,axiom,
    ! [Y: real] : aa_int_real(power_int_real(Y),aa_int_int(uminus_uminus_int,one_one_int)) = aa_real_real(inverse_inverse_real,Y) ).

% power_int_minus1_right
tff(fact_5272_power__int__minus1__right,axiom,
    ! [Y: complex] : aa_int_complex(power_int_complex(Y),aa_int_int(uminus_uminus_int,one_one_int)) = aa_complex_complex(invers8013647133539491842omplex,Y) ).

% power_int_minus1_right
tff(fact_5273_power__int__minus1__right,axiom,
    ! [Y: rat] : aa_int_rat(power_int_rat(Y),aa_int_int(uminus_uminus_int,one_one_int)) = aa_rat_rat(inverse_inverse_rat,Y) ).

% power_int_minus1_right
tff(fact_5274_round__neg__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_5275_round__neg__numeral,axiom,
    ! [N: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_5276_norm__of__int,axiom,
    ! [Z: int] : aa_real_real(real_V7735802525324610683m_real,aa_int_real(ring_1_of_int_real,Z)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Z)) ).

% norm_of_int
tff(fact_5277_norm__of__int,axiom,
    ! [Z: int] : aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(ring_17405671764205052669omplex,Z)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Z)) ).

% norm_of_int
tff(fact_5278_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(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_5279_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_5280_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_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_5281_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_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_5282_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_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_5283_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 )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),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)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5284_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5285_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5286_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))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5287_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5288_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5289_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_5290_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_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_5291_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_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_5292_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_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_5293_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_5294_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_5295_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_5296_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_dec_simps(1)
tff(fact_5297_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_5298_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_5299_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_5300_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_5301_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_inc_simps(1)
tff(fact_5302_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_5303_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_5304_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_5305_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_5306_power__int__mono__iff,axiom,
    ! [A: real,B: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(A),N)),aa_int_real(power_int_real(B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5307_power__int__mono__iff,axiom,
    ! [A: rat,B: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5308_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_5309_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_5310_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_5311_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_5312_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_5313_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(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_5314_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(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_5315_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(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_5316_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5317_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5318_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5319_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5320_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5321_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5322_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5323_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5324_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5325_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5326_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5327_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5328_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5329_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5330_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5331_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5332_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5333_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5334_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5335_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5336_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_less_floor
tff(fact_5337_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_less_floor
tff(fact_5338_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% floor_le_numeral
tff(fact_5339_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% floor_le_numeral
tff(fact_5340_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_5341_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_5342_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_le_ceiling
tff(fact_5343_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_le_ceiling
tff(fact_5344_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_5345_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_5346_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% floor_less_neg_numeral
tff(fact_5347_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% floor_less_neg_numeral
tff(fact_5348_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_5349_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_5350_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5351_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5352_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5353_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5354_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5355_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5356_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5357_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5358_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_5359_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_5360_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5361_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5362_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5363_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5364_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5365_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5366_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5367_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5368_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5369_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N) = 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,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5370_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5371_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5372_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N) = 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,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5373_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5374_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5375_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5376_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5377_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5378_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(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_5379_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_less_floor
tff(fact_5380_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_less_floor
tff(fact_5381_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% floor_le_neg_numeral
tff(fact_5382_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% floor_le_neg_numeral
tff(fact_5383_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),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_5384_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% ceiling_less_neg_numeral
tff(fact_5385_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5386_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5387_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5388_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5389_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5390_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5391_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5392_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5393_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5394_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_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,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5395_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_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,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5396_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5397_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5398_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5399_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5400_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5401_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5402_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5403_dist__norm,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V975177566351809787t_real(X),Y) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y)) ).

% dist_norm
tff(fact_5404_dist__norm,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V3694042436643373181omplex(X),Y) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y)) ).

% dist_norm
tff(fact_5405_norm__conv__dist,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,X) = aa_real_real(real_V975177566351809787t_real(X),zero_zero_real) ).

% norm_conv_dist
tff(fact_5406_norm__conv__dist,axiom,
    ! [X: complex] : aa_complex_real(real_V1022390504157884413omplex,X) = aa_complex_real(real_V3694042436643373181omplex(X),zero_zero_complex) ).

% norm_conv_dist
tff(fact_5407_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_5408_nat__numeral__as__int,axiom,
    ! [X4: num] : aa_num_nat(numeral_numeral_nat,X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).

% nat_numeral_as_int
tff(fact_5409_power__int__commutes,axiom,
    ! [X: real,N: int] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),N)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_int_real(power_int_real(X),N)) ).

% power_int_commutes
tff(fact_5410_power__int__commutes,axiom,
    ! [X: rat,N: int] : aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),N)),X) = aa_rat_rat(times_times_rat(X),aa_int_rat(power_int_rat(X),N)) ).

% power_int_commutes
tff(fact_5411_power__int__mult__distrib,axiom,
    ! [X: real,Y: real,M: int] : aa_int_real(power_int_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(Y),M)) ).

% power_int_mult_distrib
tff(fact_5412_power__int__mult__distrib,axiom,
    ! [X: rat,Y: rat,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(times_times_rat(X),Y)),M) = aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(Y),M)) ).

% power_int_mult_distrib
tff(fact_5413_power__int__divide__distrib,axiom,
    ! [X: complex,Y: complex,M: int] : aa_int_complex(power_int_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),M) = aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(power_int_complex(X),M)),aa_int_complex(power_int_complex(Y),M)) ).

% power_int_divide_distrib
tff(fact_5414_power__int__divide__distrib,axiom,
    ! [X: real,Y: real,M: int] : aa_int_real(power_int_real(aa_real_real(divide_divide_real(X),Y)),M) = aa_real_real(divide_divide_real(aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(Y),M)) ).

% power_int_divide_distrib
tff(fact_5415_power__int__divide__distrib,axiom,
    ! [X: rat,Y: rat,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(divide_divide_rat(X),Y)),M) = aa_rat_rat(divide_divide_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(Y),M)) ).

% power_int_divide_distrib
tff(fact_5416_power__int__abs,axiom,
    ! [A: real,N: int] : aa_real_real(abs_abs_real,aa_int_real(power_int_real(A),N)) = aa_int_real(power_int_real(aa_real_real(abs_abs_real,A)),N) ).

% power_int_abs
tff(fact_5417_power__int__abs,axiom,
    ! [A: rat,N: int] : aa_rat_rat(abs_abs_rat,aa_int_rat(power_int_rat(A),N)) = aa_int_rat(power_int_rat(aa_rat_rat(abs_abs_rat,A)),N) ).

% power_int_abs
tff(fact_5418_power__int__inverse,axiom,
    ! [X: real,N: int] : aa_int_real(power_int_real(aa_real_real(inverse_inverse_real,X)),N) = aa_real_real(inverse_inverse_real,aa_int_real(power_int_real(X),N)) ).

% power_int_inverse
tff(fact_5419_power__int__inverse,axiom,
    ! [X: complex,N: int] : aa_int_complex(power_int_complex(aa_complex_complex(invers8013647133539491842omplex,X)),N) = aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(power_int_complex(X),N)) ).

% power_int_inverse
tff(fact_5420_power__int__inverse,axiom,
    ! [X: rat,N: int] : aa_int_rat(power_int_rat(aa_rat_rat(inverse_inverse_rat,X)),N) = aa_rat_rat(inverse_inverse_rat,aa_int_rat(power_int_rat(X),N)) ).

% power_int_inverse
tff(fact_5421_power__int__mult,axiom,
    ! [X: real,M: int,N: int] : aa_int_real(power_int_real(X),aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = aa_int_real(power_int_real(aa_int_real(power_int_real(X),M)),N) ).

% power_int_mult
tff(fact_5422_norm__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),A)) ).

% norm_minus_commute
tff(fact_5423_norm__minus__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),A)) ).

% norm_minus_commute
tff(fact_5424_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_5425_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_5426_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_5427_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_5428_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_5429_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).

% neg_numeral_neq_numeral
tff(fact_5430_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,N) ).

% neg_numeral_neq_numeral
tff(fact_5431_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N) ).

% neg_numeral_neq_numeral
tff(fact_5432_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) != aa_num_Code_integer(numera6620942414471956472nteger,N) ).

% neg_numeral_neq_numeral
tff(fact_5433_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,N) ).

% neg_numeral_neq_numeral
tff(fact_5434_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% numeral_neq_neg_numeral
tff(fact_5435_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% numeral_neq_neg_numeral
tff(fact_5436_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% numeral_neq_neg_numeral
tff(fact_5437_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(numera6620942414471956472nteger,M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% numeral_neq_neg_numeral
tff(fact_5438_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_neq_neg_numeral
tff(fact_5439_Ints__numeral,axiom,
    ! [N: num] : member_complex(numera6690914467698888265omplex(N),ring_1_Ints_complex) ).

% Ints_numeral
tff(fact_5440_Ints__numeral,axiom,
    ! [N: num] : member_real(aa_num_real(numeral_numeral_real,N),ring_1_Ints_real) ).

% Ints_numeral
tff(fact_5441_Ints__numeral,axiom,
    ! [N: num] : member_rat(aa_num_rat(numeral_numeral_rat,N),ring_1_Ints_rat) ).

% Ints_numeral
tff(fact_5442_Ints__numeral,axiom,
    ! [N: num] : member_int(aa_num_int(numeral_numeral_int,N),ring_1_Ints_int) ).

% Ints_numeral
tff(fact_5443_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_5444_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_5445_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_5446_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)) ).

% of_int_neg_numeral
tff(fact_5447_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_5448_zero__le__power__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(power_int_real(X),N))) ) ).

% zero_le_power_int
tff(fact_5449_zero__le__power__int,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(power_int_rat(X),N))) ) ).

% zero_le_power_int
tff(fact_5450_zero__less__power__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(power_int_real(X),N))) ) ).

% zero_less_power_int
tff(fact_5451_zero__less__power__int,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(power_int_rat(X),N))) ) ).

% zero_less_power_int
tff(fact_5452_power__int__one__over,axiom,
    ! [X: complex,N: int] : aa_int_complex(power_int_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),X)),N) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_int_complex(power_int_complex(X),N)) ).

% power_int_one_over
tff(fact_5453_power__int__one__over,axiom,
    ! [X: real,N: int] : aa_int_real(power_int_real(aa_real_real(divide_divide_real(one_one_real),X)),N) = aa_real_real(divide_divide_real(one_one_real),aa_int_real(power_int_real(X),N)) ).

% power_int_one_over
tff(fact_5454_power__int__one__over,axiom,
    ! [X: rat,N: int] : aa_int_rat(power_int_rat(aa_rat_rat(divide_divide_rat(one_one_rat),X)),N) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_int_rat(power_int_rat(X),N)) ).

% power_int_one_over
tff(fact_5455_power__int__not__zero,axiom,
    ! [X: real,N: int] :
      ( ( ( X != zero_zero_real )
        | ( N = zero_zero_int ) )
     => ( aa_int_real(power_int_real(X),N) != zero_zero_real ) ) ).

% power_int_not_zero
tff(fact_5456_power__int__not__zero,axiom,
    ! [X: rat,N: int] :
      ( ( ( X != zero_zero_rat )
        | ( N = zero_zero_int ) )
     => ( aa_int_rat(power_int_rat(X),N) != zero_zero_rat ) ) ).

% power_int_not_zero
tff(fact_5457_power__int__minus,axiom,
    ! [X: real,N: int] : aa_int_real(power_int_real(X),aa_int_int(uminus_uminus_int,N)) = aa_real_real(inverse_inverse_real,aa_int_real(power_int_real(X),N)) ).

% power_int_minus
tff(fact_5458_power__int__minus,axiom,
    ! [X: complex,N: int] : aa_int_complex(power_int_complex(X),aa_int_int(uminus_uminus_int,N)) = aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(power_int_complex(X),N)) ).

% power_int_minus
tff(fact_5459_power__int__minus,axiom,
    ! [X: rat,N: int] : aa_int_rat(power_int_rat(X),aa_int_int(uminus_uminus_int,N)) = aa_rat_rat(inverse_inverse_rat,aa_int_rat(power_int_rat(X),N)) ).

% power_int_minus
tff(fact_5460_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_le_numeral
tff(fact_5461_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_le_numeral
tff(fact_5462_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_5463_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_le_numeral
tff(fact_5464_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_5465_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_5466_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_5467_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_5468_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_less_numeral
tff(fact_5469_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_less_numeral
tff(fact_5470_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_5471_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_less_numeral
tff(fact_5472_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_5473_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_5474_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_5475_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_5476_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N))) ).

% one_le_numeral
tff(fact_5477_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% one_le_numeral
tff(fact_5478_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_5479_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).

% one_le_numeral
tff(fact_5480_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_5481_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_5482_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_5483_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_5484_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_5485_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% neg_numeral_le_numeral
tff(fact_5486_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_5487_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_5488_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5489_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5490_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5491_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5492_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_5493_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% zero_neq_neg_numeral
tff(fact_5494_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_5495_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% zero_neq_neg_numeral
tff(fact_5496_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% zero_neq_neg_numeral
tff(fact_5497_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_5498_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_5499_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% neg_numeral_less_numeral
tff(fact_5500_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_5501_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5502_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5503_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5504_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5505_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_5506_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_5507_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_5508_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_5509_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_5510_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5511_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_5512_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(X),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5513_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))) ).

% numeral_times_minus_swap
tff(fact_5514_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(times_times_rat(X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_5515_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5516_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5517_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5518_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5519_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5520_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_5521_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% one_neq_neg_numeral
tff(fact_5522_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_5523_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% one_neq_neg_numeral
tff(fact_5524_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% one_neq_neg_numeral
tff(fact_5525_norm__inverse,axiom,
    ! [A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_inverse
tff(fact_5526_norm__inverse,axiom,
    ! [A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_inverse
tff(fact_5527_power__int__0__left__If,axiom,
    ! [M: int] :
      ( ( ( M = zero_zero_int )
       => ( aa_int_complex(power_int_complex(zero_zero_complex),M) = one_one_complex ) )
      & ( ( M != zero_zero_int )
       => ( aa_int_complex(power_int_complex(zero_zero_complex),M) = zero_zero_complex ) ) ) ).

% power_int_0_left_If
tff(fact_5528_power__int__0__left__If,axiom,
    ! [M: int] :
      ( ( ( M = zero_zero_int )
       => ( aa_int_real(power_int_real(zero_zero_real),M) = one_one_real ) )
      & ( ( M != zero_zero_int )
       => ( aa_int_real(power_int_real(zero_zero_real),M) = zero_zero_real ) ) ) ).

% power_int_0_left_If
tff(fact_5529_power__int__0__left__If,axiom,
    ! [M: int] :
      ( ( ( M = zero_zero_int )
       => ( aa_int_rat(power_int_rat(zero_zero_rat),M) = one_one_rat ) )
      & ( ( M != zero_zero_int )
       => ( aa_int_rat(power_int_rat(zero_zero_rat),M) = zero_zero_rat ) ) ) ).

% power_int_0_left_If
tff(fact_5530_power__int__increasing,axiom,
    ! [N: int,N5: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(A),N)),aa_int_real(power_int_real(A),N5))) ) ) ).

% power_int_increasing
tff(fact_5531_power__int__increasing,axiom,
    ! [N: int,N5: int,A: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(A),N5))) ) ) ).

% power_int_increasing
tff(fact_5532_power__int__strict__increasing,axiom,
    ! [N: int,N5: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(power_int_real(A),N)),aa_int_real(power_int_real(A),N5))) ) ) ).

% power_int_strict_increasing
tff(fact_5533_power__int__strict__increasing,axiom,
    ! [N: int,N5: int,A: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(A),N5))) ) ) ).

% power_int_strict_increasing
tff(fact_5534_power__int__minus__one__minus,axiom,
    ! [N: int] : aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_int_int(uminus_uminus_int,N)) = aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% power_int_minus_one_minus
tff(fact_5535_power__int__minus__one__minus,axiom,
    ! [N: int] : aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_int_int(uminus_uminus_int,N)) = aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% power_int_minus_one_minus
tff(fact_5536_power__int__minus__one__minus,axiom,
    ! [N: int] : aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_int_int(uminus_uminus_int,N)) = aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) ).

% power_int_minus_one_minus
tff(fact_5537_power__int__diff,axiom,
    ! [X: complex,M: int,N: int] :
      ( ( ( X != zero_zero_complex )
        | ( M != N ) )
     => ( aa_int_complex(power_int_complex(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(power_int_complex(X),M)),aa_int_complex(power_int_complex(X),N)) ) ) ).

% power_int_diff
tff(fact_5538_power__int__diff,axiom,
    ! [X: real,M: int,N: int] :
      ( ( ( X != zero_zero_real )
        | ( M != N ) )
     => ( aa_int_real(power_int_real(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)) = aa_real_real(divide_divide_real(aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(X),N)) ) ) ).

% power_int_diff
tff(fact_5539_power__int__diff,axiom,
    ! [X: rat,M: int,N: int] :
      ( ( ( X != zero_zero_rat )
        | ( M != N ) )
     => ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)) = aa_rat_rat(divide_divide_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)) ) ) ).

% power_int_diff
tff(fact_5540_power__int__minus__one__diff__commute,axiom,
    ! [A: int,B: int] : aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% power_int_minus_one_diff_commute
tff(fact_5541_power__int__minus__one__diff__commute,axiom,
    ! [A: int,B: int] : aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% power_int_minus_one_diff_commute
tff(fact_5542_power__int__minus__one__diff__commute,axiom,
    ! [A: int,B: int] : aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% power_int_minus_one_diff_commute
tff(fact_5543_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_5544_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_5545_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_5546_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5547_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( aa_nat_real(power_power_real(W),N) = aa_nat_real(power_power_real(Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5548_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),N) = aa_nat_complex(power_power_complex(Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5549_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_5550_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_5551_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_5552_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_5553_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5554_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C))) ) ).

% norm_add_leD
tff(fact_5555_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_5556_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_5557_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_5558_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_5559_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_5560_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_5561_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5562_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5563_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y))) ) ).

% round_mono
tff(fact_5564_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y))))) ).

% norm_triangle_sub
tff(fact_5565_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))))) ).

% norm_triangle_sub
tff(fact_5566_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))) ).

% norm_triangle_ineq4
tff(fact_5567_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))) ).

% norm_triangle_ineq4
tff(fact_5568_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_5569_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_5570_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y))),E2)) ) ).

% norm_triangle_le_diff
tff(fact_5571_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E2)) ) ).

% norm_triangle_le_diff
tff(fact_5572_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_5573_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_5574_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_5575_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_5576_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5577_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5578_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5579_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5580_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_5581_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_5582_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_5583_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_5584_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5585_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5586_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5587_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5588_norm__diff__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5589_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B)))) ).

% norm_diff_ineq
tff(fact_5590_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_5591_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) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_5592_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_5593_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_5594_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) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_5595_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_5596_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(minus_minus_real(A),B)))) ).

% norm_triangle_ineq2
tff(fact_5597_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).

% norm_triangle_ineq2
tff(fact_5598_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5599_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_5600_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5601_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(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_5602_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5603_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% neg_one_le_numeral
tff(fact_5604_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5605_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(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_5606_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5607_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_5608_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5609_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(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_5610_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5611_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_5612_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5613_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5614_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5615_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).

% not_one_le_neg_numeral
tff(fact_5616_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5617_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5618_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(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_5619_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5620_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_5621_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5622_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(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_5623_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5624_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% neg_one_less_numeral
tff(fact_5625_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5626_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5627_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5628_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_5629_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5630_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5631_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5632_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).

% not_one_less_neg_numeral
tff(fact_5633_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5634_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5635_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5636_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5637_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5638_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_5639_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_5640_floor__le__round,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim8280529875227126926d_real,X))) ).

% floor_le_round
tff(fact_5641_floor__le__round,axiom,
    ! [X: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% floor_le_round
tff(fact_5642_ceiling__ge__round,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim7802044766580827645g_real,X))) ).

% ceiling_ge_round
tff(fact_5643_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_5644_power__int__strict__decreasing,axiom,
    ! [N: int,N5: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(power_int_real(A),N5)),aa_int_real(power_int_real(A),N))) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5645_power__int__strict__decreasing,axiom,
    ! [N: int,N5: int,A: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(A),N5)),aa_int_rat(power_int_rat(A),N))) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5646_power__int__mono,axiom,
    ! [X: real,Y: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(X),N)),aa_int_real(power_int_real(Y),N))) ) ) ) ).

% power_int_mono
tff(fact_5647_power__int__mono,axiom,
    ! [X: rat,Y: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(X),N)),aa_int_rat(power_int_rat(Y),N))) ) ) ) ).

% power_int_mono
tff(fact_5648_power__int__strict__antimono,axiom,
    ! [A: real,B: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),zero_zero_int))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(power_int_real(B),N)),aa_int_real(power_int_real(A),N))) ) ) ) ).

% power_int_strict_antimono
tff(fact_5649_power__int__strict__antimono,axiom,
    ! [A: rat,B: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),zero_zero_int))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(B),N)),aa_int_rat(power_int_rat(A),N))) ) ) ) ).

% power_int_strict_antimono
tff(fact_5650_one__le__power__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(power_int_real(X),N))) ) ) ).

% one_le_power_int
tff(fact_5651_one__le__power__int,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(power_int_rat(X),N))) ) ) ).

% one_le_power_int
tff(fact_5652_one__less__power__int,axiom,
    ! [A: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(power_int_real(A),N))) ) ) ).

% one_less_power_int
tff(fact_5653_one__less__power__int,axiom,
    ! [A: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(power_int_rat(A),N))) ) ) ).

% one_less_power_int
tff(fact_5654_power__int__add,axiom,
    ! [X: real,M: int,N: int] :
      ( ( ( X != zero_zero_real )
        | ( aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N) != zero_zero_int ) )
     => ( aa_int_real(power_int_real(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(X),N)) ) ) ).

% power_int_add
tff(fact_5655_power__int__add,axiom,
    ! [X: rat,M: int,N: int] :
      ( ( ( X != zero_zero_rat )
        | ( aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N) != zero_zero_int ) )
     => ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)) = aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)) ) ) ).

% power_int_add
tff(fact_5656_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( aa_nat_real(power_power_real(W),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5657_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5658_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5659_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_5660_norm__sgn,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_5661_norm__sgn,axiom,
    ! [X: complex] :
      ( ( ( X = zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = zero_zero_real ) )
      & ( ( X != zero_zero_complex )
       => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = one_one_real ) ) ) ).

% norm_sgn
tff(fact_5662_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_num_real(numeral_numeral_real,W)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_5663_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_5664_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_5665_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_5666_dist__of__int,axiom,
    ! [M: int,N: int] : aa_real_real(real_V975177566351809787t_real(aa_int_real(ring_1_of_int_real,M)),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N))) ).

% dist_of_int
tff(fact_5667_dist__of__int,axiom,
    ! [M: int,N: int] : aa_complex_real(real_V3694042436643373181omplex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_int_complex(ring_17405671764205052669omplex,N)) = aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N))) ).

% dist_of_int
tff(fact_5668_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)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5669_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5670_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5671_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) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5672_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5673_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5674_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_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(minus_minus_real(A),B)))) ).

% norm_triangle_ineq3
tff(fact_5675_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).

% norm_triangle_ineq3
tff(fact_5676_power__int__antimono,axiom,
    ! [A: real,B: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),zero_zero_int))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(B),N)),aa_int_real(power_int_real(A),N))) ) ) ) ).

% power_int_antimono
tff(fact_5677_power__int__antimono,axiom,
    ! [A: rat,B: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),zero_zero_int))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(B),N)),aa_int_rat(power_int_rat(A),N))) ) ) ) ).

% power_int_antimono
tff(fact_5678_power__int__strict__mono,axiom,
    ! [A: real,B: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(power_int_real(A),N)),aa_int_real(power_int_real(B),N))) ) ) ) ).

% power_int_strict_mono
tff(fact_5679_power__int__strict__mono,axiom,
    ! [A: rat,B: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(B),N))) ) ) ) ).

% power_int_strict_mono
tff(fact_5680_power__int__decreasing,axiom,
    ! [N: int,N5: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => ( ( ( A != zero_zero_real )
              | ( N5 != zero_zero_int )
              | ( N = zero_zero_int ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(A),N5)),aa_int_real(power_int_real(A),N))) ) ) ) ) ).

% power_int_decreasing
tff(fact_5681_power__int__decreasing,axiom,
    ! [N: int,N5: int,A: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => ( ( ( A != zero_zero_rat )
              | ( N5 != zero_zero_int )
              | ( N = zero_zero_int ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(A),N5)),aa_int_rat(power_int_rat(A),N))) ) ) ) ) ).

% power_int_decreasing
tff(fact_5682_power__int__le__one,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(X),N)),one_one_real)) ) ) ) ).

% power_int_le_one
tff(fact_5683_power__int__le__one,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(X),N)),one_one_rat)) ) ) ) ).

% power_int_le_one
tff(fact_5684_power__int__le__imp__le__exp,axiom,
    ! [X: real,M: int,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(X),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N)) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5685_power__int__le__imp__le__exp,axiom,
    ! [X: rat,M: int,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N)) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5686_power__int__le__imp__less__exp,axiom,
    ! [X: real,M: int,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(power_int_real(X),M)),aa_int_real(power_int_real(X),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5687_power__int__le__imp__less__exp,axiom,
    ! [X: rat,M: int,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5688_power__int__minus__mult,axiom,
    ! [X: real,N: int] :
      ( ( ( X != zero_zero_real )
        | ( N != zero_zero_int ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int))),X) = aa_int_real(power_int_real(X),N) ) ) ).

% power_int_minus_mult
tff(fact_5689_power__int__minus__mult,axiom,
    ! [X: rat,N: int] :
      ( ( ( X != zero_zero_rat )
        | ( N != zero_zero_int ) )
     => ( aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int))),X) = aa_int_rat(power_int_rat(X),N) ) ) ).

% power_int_minus_mult
tff(fact_5690_power__int__add__1_H,axiom,
    ! [X: real,M: int] :
      ( ( ( X != zero_zero_real )
        | ( M != aa_int_int(uminus_uminus_int,one_one_int) ) )
     => ( aa_int_real(power_int_real(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),one_one_int)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_int_real(power_int_real(X),M)) ) ) ).

% power_int_add_1'
tff(fact_5691_power__int__add__1_H,axiom,
    ! [X: rat,M: int] :
      ( ( ( X != zero_zero_rat )
        | ( M != aa_int_int(uminus_uminus_int,one_one_int) ) )
     => ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),one_one_int)) = aa_rat_rat(times_times_rat(X),aa_int_rat(power_int_rat(X),M)) ) ) ).

% power_int_add_1'
tff(fact_5692_power__int__add__1,axiom,
    ! [X: real,M: int] :
      ( ( ( X != zero_zero_real )
        | ( M != aa_int_int(uminus_uminus_int,one_one_int) ) )
     => ( aa_int_real(power_int_real(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),one_one_int)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(power_int_real(X),M)),X) ) ) ).

% power_int_add_1
tff(fact_5693_power__int__add__1,axiom,
    ! [X: rat,M: int] :
      ( ( ( X != zero_zero_rat )
        | ( M != aa_int_int(uminus_uminus_int,one_one_int) ) )
     => ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),one_one_int)) = aa_rat_rat(times_times_rat(aa_int_rat(power_int_rat(X),M)),X) ) ) ).

% power_int_add_1
tff(fact_5694_Cauchy__altdef,axiom,
    ! [F: fun_nat_real] :
      ( topolo4055970368930404560y_real(F)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [M7: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
             => ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V975177566351809787t_real(aa_nat_real(F,M4)),aa_nat_real(F,N4))),E3)) ) ) ) ) ).

% Cauchy_altdef
tff(fact_5695_CauchyI_H,axiom,
    ! [X7: fun_nat_real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => ? [M8: nat] :
            ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M2))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V975177566351809787t_real(aa_nat_real(X7,M2)),aa_nat_real(X7,N2))),E)) ) ) )
     => topolo4055970368930404560y_real(X7) ) ).

% CauchyI'
tff(fact_5696_dist__of__nat,axiom,
    ! [M: nat,N: nat] : aa_real_real(real_V975177566351809787t_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))) ).

% dist_of_nat
tff(fact_5697_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_5698_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_5699_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_num_real(numeral_numeral_real,W)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_5700_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_5701_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5702_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5703_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(divide_divide_real(B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5704_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5705_norm__inverse__le__norm,axiom,
    ! [R2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,X))),aa_real_real(inverse_inverse_real,R2))) ) ) ).

% norm_inverse_le_norm
tff(fact_5706_norm__inverse__le__norm,axiom,
    ! [R2: real,X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,X))),aa_real_real(inverse_inverse_real,R2))) ) ) ).

% norm_inverse_le_norm
tff(fact_5707_Cauchy__iff,axiom,
    ! [X7: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X7)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [M7: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
             => ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X7,M4)),aa_nat_complex(X7,N4)))),E3)) ) ) ) ) ).

% Cauchy_iff
tff(fact_5708_Cauchy__iff,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [M7: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
             => ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(X7,M4)),aa_nat_real(X7,N4)))),E3)) ) ) ) ) ).

% Cauchy_iff
tff(fact_5709_CauchyI,axiom,
    ! [X7: fun_nat_complex] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => ? [M8: nat] :
            ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M2))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X7,M2)),aa_nat_complex(X7,N2)))),E)) ) ) )
     => topolo6517432010174082258omplex(X7) ) ).

% CauchyI
tff(fact_5710_CauchyI,axiom,
    ! [X7: fun_nat_real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => ? [M8: nat] :
            ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M2))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(X7,M2)),aa_nat_real(X7,N2)))),E)) ) ) )
     => topolo4055970368930404560y_real(X7) ) ).

% CauchyI
tff(fact_5711_CauchyD,axiom,
    ! [X7: fun_nat_complex,E2: real] :
      ( topolo6517432010174082258omplex(X7)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ? [M9: nat] :
          ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M5))
           => ! [N6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X7,M5)),aa_nat_complex(X7,N6)))),E2)) ) ) ) ) ).

% CauchyD
tff(fact_5712_CauchyD,axiom,
    ! [X7: fun_nat_real,E2: real] :
      ( topolo4055970368930404560y_real(X7)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ? [M9: nat] :
          ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M5))
           => ! [N6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(X7,M5)),aa_nat_real(X7,N6)))),E2)) ) ) ) ) ).

% CauchyD
tff(fact_5713_power__int__def,axiom,
    ! [N: int,X: real] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( aa_int_real(power_int_real(X),N) = aa_nat_real(power_power_real(X),aa_int_nat(nat2,N)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( aa_int_real(power_int_real(X),N) = aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N))) ) ) ) ).

% power_int_def
tff(fact_5714_power__int__def,axiom,
    ! [N: int,X: complex] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( aa_int_complex(power_int_complex(X),N) = aa_nat_complex(power_power_complex(X),aa_int_nat(nat2,N)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( aa_int_complex(power_int_complex(X),N) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N))) ) ) ) ).

% power_int_def
tff(fact_5715_power__int__def,axiom,
    ! [N: int,X: rat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( aa_int_rat(power_int_rat(X),N) = aa_nat_rat(power_power_rat(X),aa_int_nat(nat2,N)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( aa_int_rat(power_int_rat(X),N) = aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N))) ) ) ) ).

% power_int_def
tff(fact_5716_power__int__numeral__neg__numeral,axiom,
    ! [M: num,N: num] : aa_int_real(power_int_real(aa_num_real(numeral_numeral_real,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,aa_num_num(pow(M),N))) ).

% power_int_numeral_neg_numeral
tff(fact_5717_power__int__numeral__neg__numeral,axiom,
    ! [M: num,N: num] : aa_int_rat(power_int_rat(aa_num_rat(numeral_numeral_rat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(pow(M),N))) ).

% power_int_numeral_neg_numeral
tff(fact_5718_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_5719_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(power_power_real(Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_5720_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_5721_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_5722_norm__of__real__add1,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_5723_norm__of__real__add1,axiom,
    ! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_5724_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(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,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_5725_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_5726_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_5727_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_5728_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_5729_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_5730_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_5731_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_5732_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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))) ) ) ).

% member_bound
tff(fact_5733_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_5734_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_5735_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_real(numeral_numeral_real,N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_5736_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_5737_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_5738_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_5739_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_5740_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_5741_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_5742_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_5743_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_5744_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_5745_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = zero_zero_complex )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_5746_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_5747_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_5748_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_5749_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_5750_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_5751_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = one_one_complex )
    <=> ( X = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_5752_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_5753_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_5754_of__real__minus,axiom,
    ! [X: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(real_V1803761363581548252l_real,X)) ).

% of_real_minus
tff(fact_5755_of__real__minus,axiom,
    ! [X: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_real_complex(real_V4546457046886955230omplex,X)) ).

% of_real_minus
tff(fact_5756_minus__of__real__eq__of__real__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(real_V1803761363581548252l_real,X)) = aa_real_real(real_V1803761363581548252l_real,Y) )
    <=> ( aa_real_real(uminus_uminus_real,X) = Y ) ) ).

% minus_of_real_eq_of_real_iff
tff(fact_5757_minus__of__real__eq__of__real__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_real_complex(real_V4546457046886955230omplex,X)) = aa_real_complex(real_V4546457046886955230omplex,Y) )
    <=> ( aa_real_real(uminus_uminus_real,X) = Y ) ) ).

% minus_of_real_eq_of_real_iff
tff(fact_5758_of__real__eq__minus__of__real__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = aa_real_real(uminus_uminus_real,aa_real_real(real_V1803761363581548252l_real,Y)) )
    <=> ( X = aa_real_real(uminus_uminus_real,Y) ) ) ).

% of_real_eq_minus_of_real_iff
tff(fact_5759_of__real__eq__minus__of__real__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = aa_complex_complex(uminus1482373934393186551omplex,aa_real_complex(real_V4546457046886955230omplex,Y)) )
    <=> ( X = aa_real_real(uminus_uminus_real,Y) ) ) ).

% of_real_eq_minus_of_real_iff
tff(fact_5760_of__real__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_diff
tff(fact_5761_of__real__diff,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(minus_minus_real(X),Y)) = aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_diff
tff(fact_5762_of__real__of__nat__eq,axiom,
    ! [N: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ).

% of_real_of_nat_eq
tff(fact_5763_of__real__of__nat__eq,axiom,
    ! [N: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% of_real_of_nat_eq
tff(fact_5764_num__double,axiom,
    ! [N: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_5765_of__real__inverse,axiom,
    ! [X: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V1803761363581548252l_real,X)) ).

% of_real_inverse
tff(fact_5766_of__real__inverse,axiom,
    ! [X: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(inverse_inverse_real,X)) = aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(real_V4546457046886955230omplex,X)) ).

% of_real_inverse
tff(fact_5767_of__real__of__int__eq,axiom,
    ! [Z: int] : aa_real_real(real_V1803761363581548252l_real,aa_int_real(ring_1_of_int_real,Z)) = aa_int_real(ring_1_of_int_real,Z) ).

% of_real_of_int_eq
tff(fact_5768_of__real__of__int__eq,axiom,
    ! [Z: int] : aa_real_complex(real_V4546457046886955230omplex,aa_int_real(ring_1_of_int_real,Z)) = aa_int_complex(ring_17405671764205052669omplex,Z) ).

% of_real_of_int_eq
tff(fact_5769_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5770_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5771_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5772_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5773_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5774_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5775_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5776_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5777_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5778_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5779_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% Suc_numeral
tff(fact_5780_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5781_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5782_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5783_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5784_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_5785_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5786_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5787_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5788_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_5789_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_5790_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_5791_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_5792_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_5793_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_5794_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_5795_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_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_5796_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_5797_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_5798_one__mod__two__eq__one,axiom,
    aa_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_5799_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_5800_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_5801_one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ).

% one_mod_two_eq_one
tff(fact_5802_bits__one__mod__two__eq__one,axiom,
    aa_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_5803_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_5804_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_5805_bits__one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ).

% bits_one_mod_two_eq_one
tff(fact_5806_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_5807_power2__minus,axiom,
    ! [A: real] : aa_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5808_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_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_5809_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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_5810_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_5811_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_5812_add__2__eq__Suc_H,axiom,
    ! [N: nat] : 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_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_5813_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_5814_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_5815_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_5816_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_5817_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_5818_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_5819_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_5820_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_5821_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_5822_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_5823_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_5824_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_5825_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_5826_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_5827_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_5828_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_5829_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5830_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5831_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5832_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_5833_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5834_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5835_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5836_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_5837_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_5838_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_5839_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_5840_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_5841_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_5842_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_5843_add__neg__numeral__special_I6_J,axiom,
    ! [M: 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,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_5844_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_5845_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_5846_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_5847_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_5848_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_5849_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_5850_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_5851_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_5852_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(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))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_5853_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_5854_bits__1__div__2,axiom,
    aa_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_5855_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_5856_one__div__two__eq__zero,axiom,
    aa_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_5857_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5858_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5859_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5860_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),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))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5861_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_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_5862_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5863_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_5864_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_5865_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_5866_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5867_add__neg__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5868_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_5869_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_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_5870_zero__less__power2,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5871_zero__less__power2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_5872_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_5873_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_5874_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5875_diff__numeral__special_I10_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5876_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5877_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_5878_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_5879_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5880_diff__numeral__special_I11_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5881_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5882_minus__1__div__2__eq,axiom,
    aa_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_5883_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_5884_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_5885_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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 )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_5886_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_5887_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_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_5888_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_5889_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_5890_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5891_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_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_5892_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_5893_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_5894_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5895_bits__minus__1__mod__2__eq,axiom,
    aa_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_5896_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5897_minus__1__mod__2__eq,axiom,
    aa_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_5898_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5899_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: 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))),N)) = 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))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5900_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] : aa_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))),N)) = aa_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))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5901_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_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))),N)) = 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))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5902_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5903_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5904_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_5905_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_5906_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_5907_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_5908_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_5909_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_5910_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_5911_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_5912_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_5913_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_5914_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_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_5915_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_5916_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_5917_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_5918_power__minus1__even,axiom,
    ! [N: 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))),N)) = one_one_int ).

% power_minus1_even
tff(fact_5919_power__minus1__even,axiom,
    ! [N: nat] : aa_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))),N)) = one_one_real ).

% power_minus1_even
tff(fact_5920_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_complex ).

% power_minus1_even
tff(fact_5921_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_5922_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_5923_one__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_5924_one__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_5925_floor__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_5926_floor__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_5927_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5928_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_5929_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_5930_square__powr__half,axiom,
    ! [X: real] : aa_real_real(powr_real(aa_nat_real(power_power_real(X),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,X) ).

% square_powr_half
tff(fact_5931__C3_Ohyps_C_I2_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "3.hyps"(2)
tff(fact_5932_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_5933_num__induct,axiom,
    ! [P: fun_num_bool,X: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X3: num] :
            ( pp(aa_num_bool(P,X3))
           => pp(aa_num_bool(P,aa_num_num(inc,X3))) )
       => pp(aa_num_bool(P,X)) ) ) ).

% num_induct
tff(fact_5934_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(pow(X),one) = X ).

% pow.simps(1)
tff(fact_5935_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_5936_add__One,axiom,
    ! [X: num] : aa_num_num(plus_plus_num(X),one) = aa_num_num(inc,X) ).

% add_One
tff(fact_5937_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(plus_plus_num(X),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(plus_plus_num(X),Y)) ).

% add_inc
tff(fact_5938_add__One__commute,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),N) = aa_num_num(plus_plus_num(N),one) ).

% add_One_commute
tff(fact_5939_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_5940_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(times_times_num(X),aa_num_num(inc,Y)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(X),Y)),X) ).

% mult_inc
tff(fact_5941_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_5942_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_5943_zero__power2,axiom,
    aa_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_5944_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_5945_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_5946_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_5947_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_5948_one__power2,axiom,
    aa_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_5949_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_5950_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_5951_power2__commute,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_5952_power2__commute,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_5953_power2__commute,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_5954_power2__commute,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_5955_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),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))) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5956_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5957_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(X),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))) )
    <=> ( ( X = Y )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5958_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5959_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5960_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_5961_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_5962_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: 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))),N)) ).

% double_not_eq_Suc_double
tff(fact_5963_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: 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))),N) ).

% Suc_double_not_eq_double
tff(fact_5964_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_5965_less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% less_exp
tff(fact_5966_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_5967_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5968_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5969_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5970_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),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)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_5971_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5972_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5973_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5974_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),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))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_5975_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_5976_zero__le__power2,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5977_zero__le__power2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_5978_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_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_5979_power2__less__0,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_5980_power2__less__0,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_5981_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_5982_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_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_5983_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_5984_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_5985_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_5986_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_5987_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_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_5988_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_5989_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_5990_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_5991_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_5992_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_5993_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(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_5994_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_5995_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_5996_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_5997_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_5998_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_5999_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_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_6000_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 = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6001_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 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6002_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_6003_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_6004_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_6005_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6006_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6007_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6008_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,X) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6009_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X3: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_6010_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X3),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_6011_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_6012_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_6013_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_6014_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_6015_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_6016_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_6017_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),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),N)))) ) ).

% diff_le_diff_pow
tff(fact_6018_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6019_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_6020_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_6021_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_6022_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_6023_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),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)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_6024_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_6025_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% half_gt_zero_iff
tff(fact_6026_half__gt__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6027_half__gt__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_6028_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_6029_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_6030_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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_6031_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6032_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6033_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6034_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_6035_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_6036_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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_6037_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6038_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6039_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6040_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_6041_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_6042_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),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))),X)),Y)) ).

% power2_sum
tff(fact_6043_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6044_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6045_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6046_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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))),X)),Y)) ).

% power2_sum
tff(fact_6047_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_6048_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_6049_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_6050_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_6051_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_6052_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_6053_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_6054_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),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)))))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_6055_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6056_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6057_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6058_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6059_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: 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),N)) != 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))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_6060_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: 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),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_6061_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: 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),N)) != 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_6062_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: 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),N)) != 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_6063_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_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))),N)))) ).

% zero_le_even_power'
tff(fact_6064_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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))),N)))) ).

% zero_le_even_power'
tff(fact_6065_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(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))),N)))) ).

% zero_le_even_power'
tff(fact_6066_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: 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))),N) != 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,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_6067_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != 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,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_6068_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_le_1
tff(fact_6069_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_6070_abs__square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_6071_abs__square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_6072_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_less_1
tff(fact_6073_abs__square__less__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_6074_abs__square__less__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_6075_abs__square__less__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_6076_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: 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))),N)) = 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),N))) ).

% div_exp_eq
tff(fact_6077_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(divide_divide_int(aa_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))),N)) = aa_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),N))) ).

% div_exp_eq
tff(fact_6078_minus__power__mult__self,axiom,
    ! [A: int,N: 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)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = 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))),N)) ).

% minus_power_mult_self
tff(fact_6079_minus__power__mult__self,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N)) = aa_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))),N)) ).

% minus_power_mult_self
tff(fact_6080_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)) = 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))),N)) ).

% minus_power_mult_self
tff(fact_6081_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_6082_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_6083_power__odd__eq,axiom,
    ! [A: complex,N: 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))),N))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6084_power__odd__eq,axiom,
    ! [A: real,N: nat] : aa_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))),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6085_power__odd__eq,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6086_power__odd__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6087_power__odd__eq,axiom,
    ! [A: int,N: 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))),N))) = 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),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_6088_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_6089_square__norm__one,axiom,
    ! [X: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_6090_square__norm__one,axiom,
    ! [X: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_6091_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_6092_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_6093_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_6094_numeral__Bit0,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_Bit0
tff(fact_6095_numeral__Bit0,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_Bit0
tff(fact_6096_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_6097_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_6098_exp__half__le2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6099_power__minus__Bit0,axiom,
    ! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6100_power__minus__Bit0,axiom,
    ! [X: real,K: num] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6101_power__minus__Bit0,axiom,
    ! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6102_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6103_power__minus__Bit0,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6104_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa_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))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_6105_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X))))) ).

% exp_plus_inverse_exp
tff(fact_6106_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_6107_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_6108_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_6109_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_6110_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_6111_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_6112_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_6113_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_6114_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_6115_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_6116_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_6117_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_6118_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_6119_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_6120_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_6121_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_6122_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_6123_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_6124_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_6125_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = 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))),N) ).

% Suc_nat_number_of_add
tff(fact_6126_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).

% inverse_numeral_1
tff(fact_6127_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_6128_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_num_rat(numeral_numeral_rat,one) ).

% inverse_numeral_1
tff(fact_6129_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_6130_sum__squares__bound,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_6131_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_6132_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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(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(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6133_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6134_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B))
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6135_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),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))),X)),Y)) ).

% power2_diff
tff(fact_6136_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6137_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6138_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),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))),X)),Y)) ).

% power2_diff
tff(fact_6139_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_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(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_6140_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_6141_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_6142_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = A )
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
tff(fact_6143_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_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))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6144_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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))),N)))))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6145_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))),N)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6146_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_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))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_6147_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_6148_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(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))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_6149_power__minus1__odd,axiom,
    ! [N: 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))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_6150_power__minus1__odd,axiom,
    ! [N: nat] : aa_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))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_6151_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_6152_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_6153_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_6154_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(divide_divide_int(aa_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,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_6155_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: 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))),N))),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,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_6156_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),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,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_6157_div__exp__mod__exp__eq,axiom,
    ! [A: code_natural,N: nat,M: nat] : modulo8411746178871703098atural(divide5121882707175180666atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M)) = divide5121882707175180666atural(modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) ).

% div_exp_mod_exp_eq
tff(fact_6158_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_6159_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_6160_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_6161_exp__bound__half,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% exp_bound_half
tff(fact_6162_exp__bound__half,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% exp_bound_half
tff(fact_6163_int__bit__induct,axiom,
    ! [P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(P,zero_zero_int))
     => ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K2: int] :
              ( pp(aa_int_bool(P,K2))
             => ( ( K2 != zero_zero_int )
               => pp(aa_int_bool(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] :
                ( pp(aa_int_bool(P,K2))
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(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)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_6164_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_6165_arsinh__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),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_6166_nonzero__of__real__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V1803761363581548252l_real,X)) ) ) ).

% nonzero_of_real_inverse
tff(fact_6167_nonzero__of__real__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(inverse_inverse_real,X)) = aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).

% nonzero_of_real_inverse
tff(fact_6168_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_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(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(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6169_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6170_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B))
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6171_arcosh__def,axiom,
    ! [X: real] : aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),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_6172_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_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))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_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,N),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_6173_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6174_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6175_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_natural] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo8411746178871703098atural(times_2397367101498566445atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M)),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = times_2397367101498566445atural(modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6176_cosh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cosh_real,X) = zero_zero_real )
    <=> ( aa_nat_real(power_power_real(aa_real_real(exp_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_6177_cosh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cosh_complex,X) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_6178_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6179_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6180_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q4: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6181_cosh__field__def,axiom,
    ! [Z: real] : aa_real_real(cosh_real,Z) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_6182_cosh__field__def,axiom,
    ! [Z: complex] : aa_complex_complex(cosh_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Z)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_6183_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_6184_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_6185_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_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(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_6186_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_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(divide_divide_int(B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_6187_round__def,axiom,
    ! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_6188_round__def,axiom,
    ! [X: rat] : aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_6189_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_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(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_6190_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_6191_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_6192_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_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_6193_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_6194_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_6195_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_6196_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_6197_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_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_6198_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_6199_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_6200_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_6201_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_6202_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_6203_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6204_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6205_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_6206_real__le__abs__sinh,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_6207_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_6208_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_6209_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_6210_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_6211_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_6212_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(X,M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6213_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6214_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(modulo_modulo_int(X),M) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6215_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6216_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6217_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_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(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(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6218_norm__less__p1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),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,X))),one_one_real)))) ).

% norm_less_p1
tff(fact_6219_norm__less__p1,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex)))) ).

% norm_less_p1
tff(fact_6220_of__int__round__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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_6221_of__int__round__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_6222_of__int__round__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(X),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,X)))) ).

% of_int_round_ge
tff(fact_6223_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_ge
tff(fact_6224_of__int__round__gt,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(X),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,X)))) ).

% of_int_round_gt
tff(fact_6225_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_gt
tff(fact_6226_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_6227_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z))))) ) ).

% exp_bound_lemma
tff(fact_6228_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z))))) ) ).

% exp_bound_lemma
tff(fact_6229_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_6230_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),X)),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(X),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,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_6231_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(X),aa_nat_real(power_power_real(X),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),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6232_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = 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),X)),aa_real_real(minus_minus_real(one_one_real),X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_6233_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_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(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_6234_numeral__inc,axiom,
    ! [X: num] : numera6690914467698888265omplex(aa_num_num(inc,X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% numeral_inc
tff(fact_6235_numeral__inc,axiom,
    ! [X: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% numeral_inc
tff(fact_6236_numeral__inc,axiom,
    ! [X: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% numeral_inc
tff(fact_6237_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% numeral_inc
tff(fact_6238_numeral__inc,axiom,
    ! [X: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% numeral_inc
tff(fact_6239_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_6240_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_6241_fact__double,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(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))),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% fact_double
tff(fact_6242_fact__double,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_double
tff(fact_6243_fact__double,axiom,
    ! [N: nat] : 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))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,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))),N))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_double
tff(fact_6244_of__int__round__abs__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X))),X))),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_6245_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_6246_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),aa_int_real(ring_1_of_int_real,N)))),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,X) = N ) ) ).

% round_unique'
tff(fact_6247_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,N)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N ) ) ).

% round_unique'
tff(fact_6248_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),X))),X))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6249_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_6250_round__altdef,axiom,
    ! [X: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim7802044766580827645g_real,X) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) ) ) ).

% round_altdef
tff(fact_6251_round__altdef,axiom,
    ! [X: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim2889992004027027881ng_rat,X) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) ) ) ).

% round_altdef
tff(fact_6252_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_6253_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_6254_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6255_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_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(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(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6256_pochhammer__double,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_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))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_6257_pochhammer__double,axiom,
    ! [Z: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_real(comm_s7457072308508201937r_real(Z),N))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_6258_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_6259_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(X),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)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X) = Y ) ) ) ).

% round_unique
tff(fact_6260_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_6261_norm__of__real__diff,axiom,
    ! [B: real,A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(real_V1803761363581548252l_real,B)),aa_real_real(real_V1803761363581548252l_real,A)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)))) ).

% norm_of_real_diff
tff(fact_6262_norm__of__real__diff,axiom,
    ! [B: real,A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,B)),aa_real_complex(real_V4546457046886955230omplex,A)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)))) ).

% norm_of_real_diff
tff(fact_6263_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X),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),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6264_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6265_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6266_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_6267_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_6268_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(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,N),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_6269_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6270_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(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,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_6271_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( ( 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))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_6272_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_6273_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_6274_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => ( 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))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_6275_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
     => ( 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))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_6276_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),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_6277_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6278_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_6279_bit__split__inv,axiom,
    ! [X: 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,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_6280_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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)) ).

% high_def
tff(fact_6281_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_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)) ).

% low_def
tff(fact_6282_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_6283_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_6284_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_6285_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_6286_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6287_size__neq__size__imp__neq,axiom,
    ! [X: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X) != size_size_list_bool(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6288_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6289_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6290_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6291_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_6292_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P,Ys)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_6293_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys)),aa_list_nat_nat(size_size_list_nat,Xs2)))
             => pp(aa_list_nat_bool(P,Ys)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_6294_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P,Ys)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_6295_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% unset_bit_nat_def
tff(fact_6296_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs))) ) ).

% length_pos_if_in_set
tff(fact_6297_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_6298_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_6299_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_6300_length__pos__if__in__set,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_6301_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ) ).

% length_pos_if_in_set
tff(fact_6302_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_6303_card__length,axiom,
    ! [Xs: list_complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(set_complex2(Xs))),size_s3451745648224563538omplex(Xs))) ).

% card_length
tff(fact_6304_card__length,axiom,
    ! [Xs: list_list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(set_list_nat2(Xs))),size_s3023201423986296836st_nat(Xs))) ).

% card_length
tff(fact_6305_card__length,axiom,
    ! [Xs: list_set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(set_set_nat2(Xs))),size_s3254054031482475050et_nat(Xs))) ).

% card_length
tff(fact_6306_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs))),size_s6755466524823107622T_VEBT(Xs))) ).

% card_length
tff(fact_6307_card__length,axiom,
    ! [Xs: list_bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(set_bool2(Xs))),size_size_list_bool(Xs))) ).

% card_length
tff(fact_6308_card__length,axiom,
    ! [Xs: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,set_nat2(Xs))),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% card_length
tff(fact_6309_card__length,axiom,
    ! [Xs: list_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(set_int2(Xs))),size_size_list_int(Xs))) ).

% card_length
tff(fact_6310_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),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_6311_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_6312_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( 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))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_6313_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( 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))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_6314_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6315_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se7083795435491715335atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se7083795435491715335atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6316_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6317_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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,N),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_6318_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6319_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se1617098188084679374atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se1617098188084679374atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6320_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6321_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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,N),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_6322_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info2,Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_6323_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6324_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se168947363167071951atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one)),bit_se168947363167071951atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6325_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6326_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),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,N),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_6327_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_6328_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_6329_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_6330_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_6331_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_6332_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_6333_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_6334_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_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,X)) ) ) ).

% log_base_10_eq1
tff(fact_6335_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,N: nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,set_complex2(Xs))
         => pp(aa_complex_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
       => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),N))) ) ) ).

% inthall
tff(fact_6336_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => pp(aa_real_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_6337_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N: nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => pp(aa_set_nat_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),N))) ) ) ).

% inthall
tff(fact_6338_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_6339_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => pp(aa_bool_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_6340_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_6341_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => pp(aa_int_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_6342_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_6343_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_6344_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_6345_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_6346_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_6347_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_6348_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_6349_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_6350_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_6351_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_6352_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_6353_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_6354_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_6355_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_6356_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_6357_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_6358_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_6359_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_6360_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_6361_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_6362_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_simps(1)
tff(fact_6363_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_6364_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6365_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6366_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6367_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6368_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),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_6369_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_6370_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_6371_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_6372_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_6373_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6374_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6375_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6376_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6377_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: 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,N)))) = 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))),N)) ).

% div_Suc_eq_div_add3
tff(fact_6378_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_6379_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_6380_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_6381_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: 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,N)))) = 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))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_6382_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6383_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_6384_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6385_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6386_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_6387_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_6388_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_6389_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_6390_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_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(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_6391_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6392_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_6393_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_6394_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(N),K))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_6395_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_6396_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_6397_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_6398_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys2,I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6399_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys2: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Ys2),I2)) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6400_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6401_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_6402_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),nth_VEBT_VEBT(Xs3,I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_6403_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),aa_nat_bool(nth_bool(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_6404_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(nth_nat(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_6405_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),aa_nat_int(nth_int(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_6406_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys2,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6407_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys2: list_bool] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I3))
            <=> pp(aa_nat_bool(nth_bool(Ys2),I3)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6408_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6409_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6410_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_6411_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% dbl_def
tff(fact_6412_dbl__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_numeral_dbl_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X) ).

% dbl_def
tff(fact_6413_dbl__def,axiom,
    ! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X) ).

% dbl_def
tff(fact_6414_inc_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_6415_inc_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit1,X)) = aa_num_num(bit0,aa_num_num(inc,X)) ).

% inc.simps(3)
tff(fact_6416_nth__mem,axiom,
    ! [N: nat,Xs: list_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => member_complex(aa_nat_complex(nth_complex(Xs),N),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_6417_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => member_real(aa_nat_real(nth_real(Xs),N),set_real2(Xs)) ) ).

% nth_mem
tff(fact_6418_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),N),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_6419_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_6420_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(aa_nat_bool(nth_bool(Xs),N),set_bool2(Xs)) ) ).

% nth_mem
tff(fact_6421_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_6422_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => member_int(aa_nat_int(nth_int(Xs),N),set_int2(Xs)) ) ).

% nth_mem
tff(fact_6423_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X3)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_6424_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( ! [X3: bool] :
            ( member_bool(X3,set_bool2(Xs))
           => pp(aa_bool_bool(P,X3)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_6425_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X3)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_6426_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X3: int] :
            ( member_int(X3,set_int2(Xs))
           => pp(aa_int_bool(P,X3)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_6427_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3451745648224563538omplex(Xs)))
          & ( aa_nat_complex(nth_complex(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6428_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6429_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6430_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6431_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I3))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_6432_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6433_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6434_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,X: complex] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3451745648224563538omplex(Xs)))
         => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),I2))) )
     => ( member_complex(X,set_complex2(Xs))
       => pp(aa_complex_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6435_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) )
     => ( member_real(X,set_real2(Xs))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6436_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),I2))) )
     => ( member_set_nat(X,set_set_nat2(Xs))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6437_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I2))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6438_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X: bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) )
     => ( member_bool(X,set_bool2(Xs))
       => pp(aa_bool_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6439_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6440_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) )
     => ( member_int(X,set_int2(Xs))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_6441_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_6442_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X5: bool] :
          ( member_bool(X5,set_bool2(Xs))
         => pp(aa_bool_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_6443_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_6444_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( member_int(X5,set_int2(Xs))
         => pp(aa_int_bool(P,X5)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_6445_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_6446_numeral__Bit1,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_Bit1
tff(fact_6447_numeral__Bit1,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_Bit1
tff(fact_6448_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_6449_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_6450_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_6451_power__minus__Bit1,axiom,
    ! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6452_power__minus__Bit1,axiom,
    ! [X: real,K: num] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6453_power__minus__Bit1,axiom,
    ! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6454_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6455_power__minus__Bit1,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6456_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6457_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6458_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q4: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6459_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_6460_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_6461_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_6462_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X22: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X22)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_6463_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_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,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6464_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: 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,Q4))) = 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,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6465_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6466_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6467_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: 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,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6468_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,N),aa_num_Code_integer(numera6620942414471956472nteger,Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6469_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: 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)))),N) = 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)),N) ).

% Suc_div_eq_add3_div
tff(fact_6470_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: 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)))),N) = 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)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_6471_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_6472_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6473_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_6474_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_6475_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_6476_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(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))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6477_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6478_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6479_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),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,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_6480_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(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))),N))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6481_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))),N))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6482_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_6483_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(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,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_6484_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(log2(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,X)) ) ) ).

% log_base_10_eq2
tff(fact_6485_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_6486_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_6487_in__children__def,axiom,
    ! [N: nat,TreeList2: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList2),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_6488_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(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_6489_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),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))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_6490_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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(bit_concat_bit(N,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_6491_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),X)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_6492_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6493_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6494_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_6495_binomial__n__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ).

% binomial_n_n
tff(fact_6496_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_6497_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_6498_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_6499_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_6500_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_6501_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6502_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_6503_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_6504_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_6505_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(N,K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_6506_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_concat_bit(N,K),L)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).

% concat_bit_negative_iff
tff(fact_6507_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_6508_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_6509_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_6510_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_6511_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_6512_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_6513_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_6514_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_6515_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_6516_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(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_6517_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(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_6518_choose__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).

% choose_one
tff(fact_6519_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(N,K),aa_int_int(bit_concat_bit(M,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N),aa_int_int(bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_6520_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6521_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_6522_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).

% Suc_times_binomial
tff(fact_6523_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_6524_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_6525_binomial__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),K) ).

% binomial_gbinomial
tff(fact_6526_binomial__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) ).

% binomial_gbinomial
tff(fact_6527_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_6528_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).

% zero_less_binomial
tff(fact_6529_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_6530_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_6531_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),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,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_6532_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_6533_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_6534_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_6535_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( 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,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_6536_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6537_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6538_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_6539_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_6540_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_6541_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_strict_mono
tff(fact_6542_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_6543_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_6544_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_6545_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_6546_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_6547_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).

% fact_binomial
tff(fact_6548_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_6549_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_6550_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_fact
tff(fact_6551_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_6552_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_6553_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_6554_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6555_arctan__lbound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6556_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6557_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_6558_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(aa_rea7340605660300375871l_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_6559_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_6560_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_6561_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6562_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(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_6563_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_6564_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_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))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6565_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6566_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6567_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6568_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_6569_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% int_dvd_int_iff
tff(fact_6570_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_6571_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_6572_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_6573_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_6574_dvd__0__right,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).

% dvd_0_right
tff(fact_6575_dvd__0__right,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),zero_zero_rat)) ).

% dvd_0_right
tff(fact_6576_dvd__0__right,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% dvd_0_right
tff(fact_6577_dvd__0__right,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).

% dvd_0_right
tff(fact_6578_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_6579_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_6580_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_6581_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_6582_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_6583_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_6584_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_6585_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_6586_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_6587_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6588_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_6589_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_6590_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6591_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6592_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y))
    <=> pp(aa_complex_bool(dvd_dvd_complex(X),Y)) ) ).

% minus_dvd_iff
tff(fact_6593_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6594_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,X)),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6595_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6596_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6597_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(dvd_dvd_complex(X),aa_complex_complex(uminus1482373934393186551omplex,Y)))
    <=> pp(aa_complex_bool(dvd_dvd_complex(X),Y)) ) ).

% dvd_minus_iff
tff(fact_6598_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6599_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_rat_rat(uminus_uminus_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6600_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% dvd_abs_iff
tff(fact_6601_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% dvd_abs_iff
tff(fact_6602_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% dvd_abs_iff
tff(fact_6603_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),aa_rat_rat(abs_abs_rat,K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% dvd_abs_iff
tff(fact_6604_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% abs_dvd_iff
tff(fact_6605_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% abs_dvd_iff
tff(fact_6606_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% abs_dvd_iff
tff(fact_6607_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,M)),K))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% abs_dvd_iff
tff(fact_6608_take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_6609_take__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_6610_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_6611_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_6612_cos__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(cos_real,X) ).

% cos_minus
tff(fact_6613_cos__minus,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(cos_complex,X) ).

% cos_minus
tff(fact_6614_sin__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_minus
tff(fact_6615_sin__minus,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) ).

% sin_minus
tff(fact_6616_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6617_cot__zero,axiom,
    aa_real_real(cot_real,zero_zero_real) = zero_zero_real ).

% cot_zero
tff(fact_6618_cot__minus,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(cot_real,X)) ).

% cot_minus
tff(fact_6619_cot__minus,axiom,
    ! [X: complex] : aa_complex_complex(cot_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(cot_complex,X)) ).

% cot_minus
tff(fact_6620_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_6621_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_6622_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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 )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_6623_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_6624_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_6625_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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 )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_6626_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6627_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6628_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6629_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6630_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_6631_unit__prod,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ).

% unit_prod
tff(fact_6632_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6633_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(times_times_rat(C),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6634_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6635_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6636_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6637_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(C),A)),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6638_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6639_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6640_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6641_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_6642_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6643_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_6644_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat)) ) ) ).

% unit_div
tff(fact_6645_unit__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int)) ) ) ).

% unit_div
tff(fact_6646_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6647_unit__div__1__unit,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_6648_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6649_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_6650_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6651_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_add
tff(fact_6652_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_diff
tff(fact_6653_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_6654_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6655_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_6656_dvd__imp__mod__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( dvd_dvd_Code_natural(A,B)
     => ( modulo8411746178871703098atural(B,A) = zero_z2226904508553997617atural ) ) ).

% dvd_imp_mod_0
tff(fact_6657_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_6658_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_6659_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_6660_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_6661_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_6662_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_6663_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_6664_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_6665_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_6666_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_abs_dvd_iff
tff(fact_6667_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_6668_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_6669_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_6670_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_6671_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_concat_bit(N,K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(N),K) ).

% concat_bit_of_zero_2
tff(fact_6672_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6673_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% unit_mult_div_div
tff(fact_6674_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6675_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_6676_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_6677_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_6678_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_6679_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_6680_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_6681_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_6682_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_6683_sin__of__real__pi,axiom,
    aa_complex_complex(sin_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_6684_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_6685_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6686_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6687_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_real(ring_1_of_int_real,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6688_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6689_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_rat(ring_1_of_int_rat,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6690_cos__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi2
tff(fact_6691_cos__periodic__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi
tff(fact_6692_sin__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi2
tff(fact_6693_sin__periodic__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi
tff(fact_6694_cos__minus__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_minus_pi
tff(fact_6695_cos__pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_pi_minus
tff(fact_6696_sin__minus__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_minus_pi
tff(fact_6697_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_6698_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_6699_even__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_6700_even__add,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_6701_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ).

% power_minus_odd
tff(fact_6702_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(A),N)) ) ) ).

% power_minus_odd
tff(fact_6703_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ).

% power_minus_odd
tff(fact_6704_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_minus_odd
tff(fact_6705_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_minus_odd
tff(fact_6706_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6707_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(power_power_real(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6708_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6709_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6710_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6711_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),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)))) ) ) ).

% odd_Suc_div_two
tff(fact_6712_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),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))) ) ) ).

% even_Suc_div_two
tff(fact_6713_power__int__minus__left__odd,axiom,
    ! [N: int,A: real] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_int_real(power_int_real(A),N)) ) ) ).

% power_int_minus_left_odd
tff(fact_6714_power__int__minus__left__odd,axiom,
    ! [N: int,A: complex] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(power_int_complex(A),N)) ) ) ).

% power_int_minus_left_odd
tff(fact_6715_power__int__minus__left__odd,axiom,
    ! [N: int,A: rat] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(power_int_rat(A),N)) ) ) ).

% power_int_minus_left_odd
tff(fact_6716_power__int__minus__left__even,axiom,
    ! [N: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,A)),N) = aa_int_real(power_int_real(A),N) ) ) ).

% power_int_minus_left_even
tff(fact_6717_power__int__minus__left__even,axiom,
    ! [N: int,A: complex] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_int_complex(power_int_complex(A),N) ) ) ).

% power_int_minus_left_even
tff(fact_6718_power__int__minus__left__even,axiom,
    ! [N: int,A: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_int_rat(power_int_rat(A),N) ) ) ).

% power_int_minus_left_even
tff(fact_6719_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_6720_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_6721_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_6722_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_6723_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_npi_int
tff(fact_6724_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6725_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6726_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6727_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_6728_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_6729_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_6730_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6731_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6732_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6733_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_6734_even__plus__one__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_6735_even__diff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_6736_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_6737_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_6738_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_6739_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_6740_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_6741_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_6742_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_6743_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_6744_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_6745_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_6746_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_6747_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_6748_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_6749_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_6750_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_6751_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_6752_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N))) ) ) ).

% even_diff_nat
tff(fact_6753_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6754_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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 )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6755_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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 )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6756_even__succ__div__2,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6757_even__succ__div__2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6758_even__succ__div__two,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6759_even__succ__div__two,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6760_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6761_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_6762_even__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_6763_even__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_6764_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_6765_take__bit__Suc__0,axiom,
    ! [A: code_natural] : bit_se569199155075624693atural(aa_nat_nat(suc,zero_zero_nat),A) = modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_6766_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_6767_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_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))) ).

% take_bit_Suc_0
tff(fact_6768_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( 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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_6769_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_6770_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_6771_cos__npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi
tff(fact_6772_cos__npi2,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi2
tff(fact_6773_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6774_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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(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_6775_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6776_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6777_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6778_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6779_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6780_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_6781_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_6782_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_6783_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_6784_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_6785_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_6786_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_6787_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_6788_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_6789_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_6790_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_6791_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).

% cos_int_2pin
tff(fact_6792_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = one_one_real ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% cos_npi_int
tff(fact_6793_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_6794_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_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))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_6795_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_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_6796_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_6797_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_6798_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_6799_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_trans
tff(fact_6800_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_trans
tff(fact_6801_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_6802_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_6803_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] : bit_se1745604003318907178nteger(N,aa_nat_Code_integer(semiri4939895301339042750nteger,M)) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).

% take_bit_of_nat
tff(fact_6804_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).

% take_bit_of_nat
tff(fact_6805_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).

% take_bit_of_nat
tff(fact_6806_take__bit__of__int,axiom,
    ! [N: nat,K: int] : bit_se1745604003318907178nteger(N,aa_int_Code_integer(ring_18347121197199848620nteger,K)) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_of_int
tff(fact_6807_take__bit__of__int,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_of_int
tff(fact_6808_sin__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_6809_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_6810_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_6811_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_6812_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(bit_se2923211474154528505it_int(N),B))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_6813_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_6814_dvd__0__left,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_6815_dvd__0__left,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_6816_dvd__0__left,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_6817_dvd__0__left,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_6818_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_6819_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_6820_dvd__triv__right,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))) ).

% dvd_triv_right
tff(fact_6821_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A))) ).

% dvd_triv_right
tff(fact_6822_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ).

% dvd_triv_right
tff(fact_6823_dvd__triv__right,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))) ).

% dvd_triv_right
tff(fact_6824_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C)) ) ).

% dvd_mult_right
tff(fact_6825_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C)) ) ).

% dvd_mult_right
tff(fact_6826_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ).

% dvd_mult_right
tff(fact_6827_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ).

% dvd_mult_right
tff(fact_6828_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_6829_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_6830_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6831_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_6832_dvd__triv__left,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ).

% dvd_triv_left
tff(fact_6833_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B))) ).

% dvd_triv_left
tff(fact_6834_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ).

% dvd_triv_left
tff(fact_6835_dvd__triv__left,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ).

% dvd_triv_left
tff(fact_6836_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).

% dvd_mult_left
tff(fact_6837_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ).

% dvd_mult_left
tff(fact_6838_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).

% dvd_mult_left
tff(fact_6839_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).

% dvd_mult_left
tff(fact_6840_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult2
tff(fact_6841_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult2
tff(fact_6842_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult2
tff(fact_6843_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ).

% dvd_mult2
tff(fact_6844_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ? [B6: nat,C4: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B6),B))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C4),C)) ) ) ).

% division_decomp
tff(fact_6845_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ? [B6: int,C4: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C4) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B6),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C4),C)) ) ) ).

% division_decomp
tff(fact_6846_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult
tff(fact_6847_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult
tff(fact_6848_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult
tff(fact_6849_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ).

% dvd_mult
tff(fact_6850_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y3) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X3),A))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_6851_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P3),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ~ ! [X3: int,Y3: int] :
            ( ( P3 = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y3) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X3),A))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_6852_dvd__def,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_6853_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_6854_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6855_dvd__def,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6856_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_6857_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_6858_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_6859_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_6860_dvdE,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_6861_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ~ ! [K2: rat] : A != aa_rat_rat(times_times_rat(B),K2) ) ).

% dvdE
tff(fact_6862_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_6863_dvdE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_6864_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_6865_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_6866_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_6867_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% unit_imp_dvd
tff(fact_6868_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% unit_imp_dvd
tff(fact_6869_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(dvd_dvd_complex(one_one_complex),A)) ).

% one_dvd
tff(fact_6870_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_6871_one__dvd,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A)) ).

% one_dvd
tff(fact_6872_one__dvd,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).

% one_dvd
tff(fact_6873_one__dvd,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).

% one_dvd
tff(fact_6874_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ) ).

% dvd_add
tff(fact_6875_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ) ).

% dvd_add
tff(fact_6876_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ) ).

% dvd_add
tff(fact_6877_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ) ).

% dvd_add
tff(fact_6878_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6879_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6880_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6881_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6882_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6883_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6884_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6885_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6886_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(minus_minus_real(Y),Z))) ) ) ).

% dvd_diff
tff(fact_6887_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_rat_rat(minus_minus_rat(Y),Z))) ) ) ).

% dvd_diff
tff(fact_6888_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z))) ) ) ).

% dvd_diff
tff(fact_6889_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% dvd_diff_commute
tff(fact_6890_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6891_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),D)),aa_int_int(divide_divide_int(B),D)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_div_div_same
tff(fact_6892_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( pp(aa_complex_bool(dvd_dvd_complex(C),A))
       => ( pp(aa_complex_bool(dvd_dvd_complex(C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6893_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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6894_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6895_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) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6896_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6897_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(dvd_dvd_complex(C),A))
     => ( pp(aa_complex_bool(dvd_dvd_complex(C),B))
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6898_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6899_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6900_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6901_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6902_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_6903_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_6904_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_6905_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_6906_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_6907_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_6908_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6909_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6910_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6911_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6912_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6913_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6914_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6915_dvd__mod__imp__dvd,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( dvd_dvd_Code_natural(C,modulo8411746178871703098atural(A,B))
     => ( dvd_dvd_Code_natural(C,B)
       => dvd_dvd_Code_natural(C,A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6916_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_6917_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_6918_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_6919_dvd__mod__iff,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( dvd_dvd_Code_natural(C,B)
     => ( dvd_dvd_Code_natural(C,modulo8411746178871703098atural(A,B))
      <=> dvd_dvd_Code_natural(C,A) ) ) ).

% dvd_mod_iff
tff(fact_6920_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_6921_uminus__dvd__conv_I2_J,axiom,
    ! [D: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(uminus_uminus_int,T))) ) ).

% uminus_dvd_conv(2)
tff(fact_6922_uminus__dvd__conv_I1_J,axiom,
    ! [D: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,D)),T)) ) ).

% uminus_dvd_conv(1)
tff(fact_6923_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_6924_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_6925_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_6926_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_6927_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6928_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6929_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_6930_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_6931_sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real)) ).

% sin_le_one
tff(fact_6932_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_6933_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N),K))) ).

% take_bit_nonnegative
tff(fact_6934_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_6935_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_6936_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(bit_ri631733984087533419it_int(N),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6937_cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real)) ).

% cos_le_one
tff(fact_6938_not__is__unit__0,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).

% not_is_unit_0
tff(fact_6939_not__is__unit__0,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).

% not_is_unit_0
tff(fact_6940_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% pinf(9)
tff(fact_6941_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))) ) ) ).

% pinf(9)
tff(fact_6942_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).

% pinf(9)
tff(fact_6943_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).

% pinf(9)
tff(fact_6944_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% pinf(10)
tff(fact_6945_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))) ) ) ).

% pinf(10)
tff(fact_6946_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X4))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).

% pinf(10)
tff(fact_6947_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X4))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).

% pinf(10)
tff(fact_6948_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% minf(9)
tff(fact_6949_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))) ) ) ).

% minf(9)
tff(fact_6950_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).

% minf(9)
tff(fact_6951_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).

% minf(9)
tff(fact_6952_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).

% minf(10)
tff(fact_6953_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))) ) ) ).

% minf(10)
tff(fact_6954_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).

% minf(10)
tff(fact_6955_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).

% minf(10)
tff(fact_6956_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(dvd_dvd_complex(B),A))
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6957_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6958_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6959_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6960_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6961_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_6962_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_6963_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_6964_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_6965_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_6966_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_6967_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6968_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6969_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6970_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6971_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6972_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6973_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6974_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6975_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6976_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),C))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_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_6977_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))) ) ).

% dvd_mult_imp_div
tff(fact_6978_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))) ) ).

% dvd_mult_imp_div
tff(fact_6979_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6980_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_6981_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6982_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_6983_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6984_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_6985_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6986_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_6987_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_6988_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_6989_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_6990_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_6991_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6992_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_6993_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6994_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6995_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6996_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6997_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_div_neg
tff(fact_6998_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6999_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(dvd_dvd_complex(B),A))
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_7000_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_7001_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_7002_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_neg_div
tff(fact_7003_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7004_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(dvd_dvd_complex(B),A))
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_7005_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_7006_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_7007_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_7008_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 )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_7009_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_7010_mod__0__imp__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural )
     => dvd_dvd_Code_natural(B,A) ) ).

% mod_0_imp_dvd
tff(fact_7011_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_7012_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7013_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_7014_dvd__eq__mod__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( dvd_dvd_Code_natural(A,B)
    <=> ( modulo8411746178871703098atural(B,A) = zero_z2226904508553997617atural ) ) ).

% dvd_eq_mod_eq_0
tff(fact_7015_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_7016_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 )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_7017_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_7018_mod__eq__0__iff__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural )
    <=> dvd_dvd_Code_natural(B,A) ) ).

% mod_eq_0_iff_dvd
tff(fact_7019_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ) ).

% mod_eq_dvd_iff
tff(fact_7020_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))) ) ).

% mod_eq_dvd_iff
tff(fact_7021_dvd__minus__mod,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)))) ).

% dvd_minus_mod
tff(fact_7022_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7023_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),modulo364778990260209775nteger(A,B)))) ).

% dvd_minus_mod
tff(fact_7024_dvd__minus__mod,axiom,
    ! [B: code_natural,A: code_natural] : dvd_dvd_Code_natural(B,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),modulo8411746178871703098atural(A,B))) ).

% dvd_minus_mod
tff(fact_7025_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_7026_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_7027_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_7028_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_7029_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_7030_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_7031_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_7032_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_7033_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(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_7034_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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),N)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_7035_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),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),Y)),D) )
            | ( 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),Y)),D) ) )
         => ? [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(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),X3) = 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)),Y3)),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)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_7036_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X3: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_7037_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X3: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D4 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_7038_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_7039_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(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,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T))) ) ) ).

% zdvd_period
tff(fact_7040_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_7041_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_7042_cos__int__times__real,axiom,
    ! [M: int,X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% cos_int_times_real
tff(fact_7043_cos__int__times__real,axiom,
    ! [M: int,X: real] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% cos_int_times_real
tff(fact_7044_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real)) ).

% sin_cos_le1
tff(fact_7045_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_7046_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_7047_sin__int__times__real,axiom,
    ! [M: int,X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% sin_int_times_real
tff(fact_7048_sin__int__times__real,axiom,
    ! [M: int,X: real] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% sin_int_times_real
tff(fact_7049_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_7050_sin__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7051_sin__squared__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7052_cos__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7053_cos__squared__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7054_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri631733984087533419it_int(N),A)) = aa_int_int(bit_se2923211474154528505it_int(M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_7055_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_7056_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_7057_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_7058_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X5: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X5)))
    <=> ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),zero_zero_real)))
          & pp(aa_real_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_7059_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X5: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X5)))
    <=> ? [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),zero_zero_rat)))
          & pp(aa_rat_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_7060_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X5: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X5)))
    <=> ? [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),zero_zero_nat)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_7061_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X5: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X5)))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),zero_zero_int)))
          & pp(aa_int_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_7062_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7063_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_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_7064_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_7065_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_7066_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_7067_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_7068_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7069_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
       => ( ( aa_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_7070_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X))) ).

% sin_ge_minus_one
tff(fact_7071_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7072_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_7073_inf__period_I4_J,axiom,
    ! [D: real,D6: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D6))
     => ! [X4: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D6))),T))) ) ) ).

% inf_period(4)
tff(fact_7074_inf__period_I4_J,axiom,
    ! [D: rat,D6: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D6))
     => ! [X4: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K4),D6))),T))) ) ) ).

% inf_period(4)
tff(fact_7075_inf__period_I4_J,axiom,
    ! [D: int,D6: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D6))
     => ! [X4: int,K4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D6))),T))) ) ) ).

% inf_period(4)
tff(fact_7076_inf__period_I3_J,axiom,
    ! [D: real,D6: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D6))
     => ! [X4: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D6))),T))) ) ) ).

% inf_period(3)
tff(fact_7077_inf__period_I3_J,axiom,
    ! [D: rat,D6: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D6))
     => ! [X4: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K4),D6))),T))) ) ) ).

% inf_period(3)
tff(fact_7078_inf__period_I3_J,axiom,
    ! [D: int,D6: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D6))
     => ! [X4: int,K4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(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,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D6))),T))) ) ) ).

% inf_period(3)
tff(fact_7079_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7080_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_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_7081_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7082_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( A = aa_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_7083_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7084_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_7085_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7086_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_7087_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7088_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_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_7089_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7090_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_7091_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X))) ).

% cos_ge_minus_one
tff(fact_7092_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),one_one_real)) ).

% abs_sin_le_one
tff(fact_7093_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_7094_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_7095_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7096_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_7097_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( dvd_dvd_Code_natural(B,one_one_Code_natural)
     => ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
tff(fact_7098_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_7099_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_7100_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X))),one_one_real)) ).

% abs_cos_le_one
tff(fact_7101_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7102_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_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_7103_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7104_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_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_7105_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7106_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_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_7107_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7108_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_7109_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7110_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_7111_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7112_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_7113_cos__double,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7114_cos__double,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7115_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_7116_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7117_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7118_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7119_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_7120_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_7121_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_7122_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X3: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),B))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_7123_sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(sin_real,X) = 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)))),X)) ).

% sin_cos_eq
tff(fact_7124_sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,X) = aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X)) ).

% sin_cos_eq
tff(fact_7125_cos__sin__eq,axiom,
    ! [X: real] : aa_real_real(cos_real,X) = 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)))),X)) ).

% cos_sin_eq
tff(fact_7126_cos__sin__eq,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,X) = aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X)) ).

% cos_sin_eq
tff(fact_7127_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_7128_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_7129_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_7130_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_7131_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_Code_integer(divide6298287555418463151nteger(aa_int_Code_integer(ring_18347121197199848620nteger,A),aa_int_Code_integer(ring_18347121197199848620nteger,B)),ring_11222124179247155820nteger) ) ).

% of_int_divide_in_Ints
tff(fact_7132_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_complex(aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(ring_17405671764205052669omplex,A)),aa_int_complex(ring_17405671764205052669omplex,B)),ring_1_Ints_complex) ) ).

% of_int_divide_in_Ints
tff(fact_7133_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_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)),ring_1_Ints_real) ) ).

% of_int_divide_in_Ints
tff(fact_7134_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_rat(aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A)),aa_int_rat(ring_1_of_int_rat,B)),ring_1_Ints_rat) ) ).

% of_int_divide_in_Ints
tff(fact_7135_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_int(aa_int_int(divide_divide_int(aa_int_int(ring_1_of_int_int,A)),aa_int_int(ring_1_of_int_int,B)),ring_1_Ints_int) ) ).

% of_int_divide_in_Ints
tff(fact_7136_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D)) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_7137_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),N))
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N),D)) = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_7138_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_7139_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_7140_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_7141_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7142_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_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_7143_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = 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))),N)),one_one_int) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_7144_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: 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 )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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))),N)),one_one_nat) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_7145_minus__sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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_7146_minus__sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_7147_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat)) ).

% even_zero
tff(fact_7148_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int)) ).

% even_zero
tff(fact_7149_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B3),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = 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),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_7150_is__unitE,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),one_one_int))
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_7151_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7152_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_7153_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7154_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_7155_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_7156_odd__one,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat)) ).

% odd_one
tff(fact_7157_odd__one,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int)) ).

% odd_one
tff(fact_7158_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7159_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_7160_even__minus,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_minus
tff(fact_7161_even__minus,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) ) ).

% even_minus
tff(fact_7162_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_7163_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_7164_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_7165_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N))) ) ).

% dvd_power
tff(fact_7166_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).

% dvd_power
tff(fact_7167_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(power_power_real(X),N))) ) ).

% dvd_power
tff(fact_7168_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))) ) ).

% dvd_power
tff(fact_7169_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(dvd_dvd_complex(X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_7170_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I3: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7171_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),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))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_7172_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).

% choose_dvd
tff(fact_7173_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% choose_dvd
tff(fact_7174_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% choose_dvd
tff(fact_7175_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% choose_dvd
tff(fact_7176_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_7177_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_7178_dvd__minus__add,axiom,
    ! [Q4: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q4)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q4)))) ) ) ) ).

% dvd_minus_add
tff(fact_7179_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),pi))
            & ( X = aa_real_real(cos_real,T2) )
            & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_7180_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_7181_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_7182_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_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,X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_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_7183_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7184_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_7185_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7186_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I3: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7187_aset_I10_J,axiom,
    ! [D: int,D6: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X4),D6)),T))) ) ) ) ).

% aset(10)
tff(fact_7188_aset_I9_J,axiom,
    ! [D: int,D6: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(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,X4),D6)),T))) ) ) ) ).

% aset(9)
tff(fact_7189_bset_I10_J,axiom,
    ! [D: int,D6: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,B2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X4),D6)),T))) ) ) ) ).

% bset(10)
tff(fact_7190_bset_I9_J,axiom,
    ! [D: int,D6: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D6))
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb: int] :
                  ( member_int(Xb,B2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(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,X4),D6)),T))) ) ) ) ).

% bset(9)
tff(fact_7191_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_7192_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_7193_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_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,X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_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_7194_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7195_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7196_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_7197_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = Acc ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_7198_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa3: nat,Xb3: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa3),Xb3),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb3),Xa3))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb3),Xa3))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa3),one_one_nat)),Xb3),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa3),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7199_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_7200_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
            & ( aa_real_real(cos_real,X3) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),pi))
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_7201_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
    <=> ( aa_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_7202_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7203_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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_7204_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_natural] :
      ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_7205_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
    <=> ( aa_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_7206_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7207_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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_7208_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_natural] :
      ( ~ dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_7209_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_7210_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(power_power_real(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_7211_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_7212_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_7213_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_nat_rat(power_power_rat(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_7214_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_7215_power__int__minus__left,axiom,
    ! [N: int,A: real] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,A)),N) = aa_int_real(power_int_real(A),N) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_real(power_int_real(aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_int_real(power_int_real(A),N)) ) ) ) ).

% power_int_minus_left
tff(fact_7216_power__int__minus__left,axiom,
    ! [N: int,A: complex] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_int_complex(power_int_complex(A),N) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_complex(power_int_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(power_int_complex(A),N)) ) ) ) ).

% power_int_minus_left
tff(fact_7217_power__int__minus__left,axiom,
    ! [N: int,A: rat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_int_rat(power_int_rat(A),N) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(power_int_rat(A),N)) ) ) ) ).

% power_int_minus_left
tff(fact_7218_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(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_7219_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7220_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(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_7221_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7222_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(M),A)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(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_7223_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7224_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_7225_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
              & ( X = aa_real_real(cos_real,T2) )
              & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7226_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
          & ( X = aa_real_real(cos_real,T2) )
          & ( Y = aa_real_real(sin_real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7227_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_7228_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_7229_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T2) )
               => ( Y != aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7230_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_7231_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_7232_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_7233_cos__integer__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_7234_oddE,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ~ ! [B3: 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))),B3)),one_one_nat) ) ).

% oddE
tff(fact_7235_oddE,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ~ ! [B3: 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))),B3)),one_one_int) ) ).

% oddE
tff(fact_7236_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7237_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_7238_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) ) ) ).

% mod2_eq_if
tff(fact_7239_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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 ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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 ) ) ) ).

% mod2_eq_if
tff(fact_7240_mod2__eq__if,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) )
      & ( ~ dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ) ).

% mod2_eq_if
tff(fact_7241_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_7242_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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 ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7243_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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 ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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_7244_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural ) )
     => ~ ( ~ dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
         => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural ) ) ) ).

% parity_cases
tff(fact_7245_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_7246_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_7247_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_7248_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_7249_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% zero_le_odd_power
tff(fact_7250_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_7251_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_le_even_power
tff(fact_7252_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_even_power
tff(fact_7253_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_even_power
tff(fact_7254_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% minus_one_power_iff
tff(fact_7255_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% minus_one_power_iff
tff(fact_7256_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% minus_one_power_iff
tff(fact_7257_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% minus_one_power_iff
tff(fact_7258_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% minus_one_power_iff
tff(fact_7259_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_7260_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7261_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_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_7262_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7263_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_7264_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_7265_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_7266_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X) = aa_real_real(sin_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7267_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7268_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7269_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_7270_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_7271_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_7272_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(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_7273_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),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))),N)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_7274_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X5: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X5)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7275_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_7276_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_7277_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_7278_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7279_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_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_7280_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_7281_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_7282_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_7283_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_7284_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_7285_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_7286_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(N,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7287_take__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se569199155075624693atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(bit_se569199155075624693atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),numera5444537566228673987atural(aa_num_num(bit0,one)))),modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7288_take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7289_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7290_cos__treble__cos,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X)) = aa_complex_complex(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(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X))) ).

% cos_treble_cos
tff(fact_7291_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),X)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_7292_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_7293_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_7294_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7295_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(sin_real,X3) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_7296_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),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))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_7297_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_7298_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_ge_zero
tff(fact_7299_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(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))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_7300_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_7301_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_7302_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X5: nat] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X5)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X5: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X5)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7303_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),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))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_7304_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_7305_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_7306_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_7307_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),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_7308_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_7309_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_7310_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_7311_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_7312_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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))),N))))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_7313_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),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_7314_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_7315_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),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_7316_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_7317_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_7318_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
      & ( ( N != zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_7319_take__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      ( ( ( N = zero_zero_nat )
       => ( bit_se569199155075624693atural(N,A) = zero_z2226904508553997617atural ) )
      & ( ( N != zero_zero_nat )
       => ( bit_se569199155075624693atural(N,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),numera5444537566228673987atural(aa_num_num(bit0,one)))),modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_7320_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_7321_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_7322_fact__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_7323_fact__code,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_7324_fact__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_7325_fact__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_7326_fact__code,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_nat)) ).

% fact_code
tff(fact_7327_sin__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(sin_coeff,X4) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(sin_coeff,X4) = aa_real_real(divide_divide_real(aa_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,X4),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) ) ) ).

% sin_coeff_def
tff(fact_7328_cos__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(cos_coeff,X4) = aa_real_real(divide_divide_real(aa_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,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
       => ( aa_nat_real(cos_coeff,X4) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_7329_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7330_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,X))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7331_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T2)),aa_real_real(sin_real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7332_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(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_7333_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7334_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7335_tan__minus,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(tan_real,X)) ).

% tan_minus
tff(fact_7336_tan__minus,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(tan_complex,X)) ).

% tan_minus
tff(fact_7337_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_7338_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7339_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_7340_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_7341_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_7342_cos__arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_7343_sin__arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_7344_norm__cos__sin,axiom,
    ! [T: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T))) = one_one_real ).

% norm_cos_sin
tff(fact_7345_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_7346_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_7347_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_7348_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_7349_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_7350_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% gcd_nat.strict_implies_order
tff(fact_7351_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_7352_gcd__nat_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
          & ( A != B ) )
        | ( A = B ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_7353_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_7354_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_7355_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
          & ( B != C ) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_7356_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
          & ( B != C ) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_7357_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( A = B ) ) ) ).

% gcd_nat.antisym
tff(fact_7358_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A))
        & ( A != A ) ) ).

% gcd_nat.irrefl
tff(fact_7359_gcd__nat_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% gcd_nat.eq_iff
tff(fact_7360_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.trans
tff(fact_7361_gcd__nat_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% gcd_nat.refl
tff(fact_7362_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
          & ( B != A ) ) ) ).

% gcd_nat.asym
tff(fact_7363_complex__minus,axiom,
    ! [A: real,B: real] : aa_complex_complex(uminus1482373934393186551omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% complex_minus
tff(fact_7364_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_7365_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_7366_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_7367_cot__altdef,axiom,
    ! [X: real] : aa_real_real(cot_real,X) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% cot_altdef
tff(fact_7368_cot__altdef,axiom,
    ! [X: complex] : aa_complex_complex(cot_complex,X) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(tan_complex,X)) ).

% cot_altdef
tff(fact_7369_tan__altdef,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(inverse_inverse_real,aa_real_real(cot_real,X)) ).

% tan_altdef
tff(fact_7370_tan__altdef,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cot_complex,X)) ).

% tan_altdef
tff(fact_7371_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_7372_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_7373_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_7374_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_7375_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_7376_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% arccos_le_mono
tff(fact_7377_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_7378_arcsin__minus,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_7379_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_7380_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_7381_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_7382_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_7383_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_7384_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_7385_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_7386_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_7387_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_7388_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_7389_arccos__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_7390_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_7391_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_7392_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7393_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_7394_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_7395_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_7396_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_7397_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_7398_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_7399_arccos__minus,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_7400_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_7401_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X3) = Y )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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 = X3 ) ) ) ).

% tan_total
tff(fact_7402_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_7403_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_7404_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_7405_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X3) = Y ) ) ).

% lemma_tan_total1
tff(fact_7406_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_7407_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_7408_tan__cot,axiom,
    ! [X: 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)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_7409_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(divide_divide_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_7410_arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_7411_arccos__minus__abs,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_7412_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7413_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != 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,X)),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,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7414_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_7415_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_7416_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_7417_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_7418_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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 )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_7419_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_7420_arctan,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7421_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7422_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),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,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7423_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7424_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7425_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7426_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != 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,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7427_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))))),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),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,Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_7428_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7429_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7430_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7431_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7432_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7433_arcsin__sin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_7434_tan__sec,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(cos_real,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_7435_tan__sec,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_7436_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).

% tan_half
tff(fact_7437_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_7438_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(arcsin,X)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_7439_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),Y))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sin_real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_7440_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7441_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_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) = aa_real_real(divide_divide_real(aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7442_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(cos_real,X) = 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(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7443_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_7444_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_7445_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_7446_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X)))
    <=> pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_7447_nat__of__bool,axiom,
    ! [P: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P) ).

% nat_of_bool
tff(fact_7448_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_7449_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_7450_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_7451_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = zero_zero_real )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_7452_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_7453_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_7454_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = zero_zero_nat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_7455_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_7456_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_7457_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_7458_of__bool__eq_I1_J,axiom,
    aa_bool_nat(zero_n2687167440665602831ol_nat,fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_7459_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_7460_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_7461_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_7462_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_7463_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = one_one_complex )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_7464_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = one_one_real )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_7465_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = one_one_rat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_7466_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_7467_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = one_one_nat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_7468_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex(fTrue) = one_one_complex ).

% of_bool_eq(2)
tff(fact_7469_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real(fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_7470_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat(fTrue) = one_one_rat ).

% of_bool_eq(2)
tff(fact_7471_of__bool__eq_I2_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_7472_of__bool__eq_I2_J,axiom,
    aa_bool_nat(zero_n2687167440665602831ol_nat,fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_7473_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_real(semiri5074537144036343181t_real,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = zero_n3304061248610475627l_real(P) ).

% of_nat_of_bool
tff(fact_7474_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = zero_n356916108424825756nteger(P) ).

% of_nat_of_bool
tff(fact_7475_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_rat(semiri681578069525770553at_rat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = zero_n2052037380579107095ol_rat(P) ).

% of_nat_of_bool
tff(fact_7476_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_int(semiri1314217659103216013at_int,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_nat_of_bool
tff(fact_7477_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_nat(semiri1316708129612266289at_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P) ).

% of_nat_of_bool
tff(fact_7478_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_7479_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_7480_abs__bool__eq,axiom,
    ! [P: bool] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real(P)) = zero_n3304061248610475627l_real(P) ).

% abs_bool_eq
tff(fact_7481_abs__bool__eq,axiom,
    ! [P: bool] : abs_abs_Code_integer(zero_n356916108424825756nteger(P)) = zero_n356916108424825756nteger(P) ).

% abs_bool_eq
tff(fact_7482_abs__bool__eq,axiom,
    ! [P: bool] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat(P)) = zero_n2052037380579107095ol_rat(P) ).

% abs_bool_eq
tff(fact_7483_abs__bool__eq,axiom,
    ! [P: bool] : aa_int_int(abs_abs_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% abs_bool_eq
tff(fact_7484_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_real(ring_1_of_int_real,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n3304061248610475627l_real(P) ).

% of_int_of_bool
tff(fact_7485_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n356916108424825756nteger(P) ).

% of_int_of_bool
tff(fact_7486_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_rat(ring_1_of_int_rat,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n2052037380579107095ol_rat(P) ).

% of_int_of_bool
tff(fact_7487_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_complex(ring_17405671764205052669omplex,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n1201886186963655149omplex(P) ).

% of_int_of_bool
tff(fact_7488_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_int(ring_1_of_int_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_int_of_bool
tff(fact_7489_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_7490_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_7491_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_7492_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_bool_nat(zero_n2687167440665602831ol_nat,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_7493_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),one_one_real))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_7494_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_7495_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_7496_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),one_one_nat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_7497_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P)) ).

% of_bool_not_iff
tff(fact_7498_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P)) = aa_real_real(minus_minus_real(one_one_real),zero_n3304061248610475627l_real(P)) ).

% of_bool_not_iff
tff(fact_7499_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat(P)) ).

% of_bool_not_iff
tff(fact_7500_of__bool__not__iff,axiom,
    ! [P: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P)) ).

% of_bool_not_iff
tff(fact_7501_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_7502_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_7503_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_7504_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% real_sqrt_le_1_iff
tff(fact_7505_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_7506_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% sgn_mult_self_eq
tff(fact_7507_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_mult_self_eq
tff(fact_7508_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% sgn_mult_self_eq
tff(fact_7509_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_mult_self_eq
tff(fact_7510_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(fequal_complex(A),zero_zero_complex))) ).

% sgn_abs
tff(fact_7511_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% sgn_abs
tff(fact_7512_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_abs
tff(fact_7513_sgn__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% sgn_abs
tff(fact_7514_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_abs
tff(fact_7515_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(fequal_complex(A),zero_zero_complex))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7516_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,abs_abs_Code_integer(A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7517_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7518_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(abs_abs_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7519_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7520_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_7521_take__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_7522_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_7523_sgn__of__nat,axiom,
    ! [N: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,N)) = zero_n3304061248610475627l_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_7524_sgn__of__nat,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_7525_sgn__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = zero_n2052037380579107095ol_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_7526_sgn__of__nat,axiom,
    ! [N: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_7527_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_int_int(divide_divide_int(aa_bool_int(zero_n2684676970156552555ol_int,B)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_7528_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_7529_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_7530_bits__1__div__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_7531_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_7532_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_7533_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(bit_se2923211474154528505it_int(M),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_7534_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_7535_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_7536_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo8411746178871703098atural(one_one_Code_natural,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = zero_n8403883297036319079atural(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_7537_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_7538_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_7539_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fconj(P),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)) ).

% of_bool_conj
tff(fact_7540_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P),Q)) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)) ).

% of_bool_conj
tff(fact_7541_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P),Q)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_conj
tff(fact_7542_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)) ).

% of_bool_conj
tff(fact_7543_real__sqrt__minus,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,X)) ).

% real_sqrt_minus
tff(fact_7544_of__bool__eq__iff,axiom,
    ! [P3: bool,Q4: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = aa_bool_int(zero_n2684676970156552555ol_int,Q4) )
    <=> ( pp(P3)
      <=> pp(Q4) ) ) ).

% of_bool_eq_iff
tff(fact_7545_of__bool__eq__iff,axiom,
    ! [P3: bool,Q4: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P3) = aa_bool_nat(zero_n2687167440665602831ol_nat,Q4) )
    <=> ( pp(P3)
      <=> pp(Q4) ) ) ).

% of_bool_eq_iff
tff(fact_7546_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_7547_real__sqrt__inverse,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ).

% real_sqrt_inverse
tff(fact_7548_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P))) ).

% zero_less_eq_of_bool
tff(fact_7549_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P))) ).

% zero_less_eq_of_bool
tff(fact_7550_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P))) ).

% zero_less_eq_of_bool
tff(fact_7551_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_bool_nat(zero_n2687167440665602831ol_nat,P))) ).

% zero_less_eq_of_bool
tff(fact_7552_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_7553_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_7554_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat)) ).

% of_bool_less_eq_one
tff(fact_7555_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_7556_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_7557_split__of__bool__asm,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_complex_bool(P,one_one_complex)) )
          | ( ~ pp(P3)
            & ~ pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_7558_split__of__bool__asm,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_real_bool(P,one_one_real)) )
          | ( ~ pp(P3)
            & ~ pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_7559_split__of__bool__asm,axiom,
    ! [P: fun_rat_bool,P3: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_rat_bool(P,one_one_rat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool_asm
tff(fact_7560_split__of__bool__asm,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_int_bool(P,one_one_int)) )
          | ( ~ pp(P3)
            & ~ pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_7561_split__of__bool__asm,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,aa_bool_nat(zero_n2687167440665602831ol_nat,P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_nat_bool(P,one_one_nat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_7562_split__of__bool,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ( ( pp(P3)
         => pp(aa_complex_bool(P,one_one_complex)) )
        & ( ~ pp(P3)
         => pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_7563_split__of__bool,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P3)))
    <=> ( ( pp(P3)
         => pp(aa_real_bool(P,one_one_real)) )
        & ( ~ pp(P3)
         => pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_7564_split__of__bool,axiom,
    ! [P: fun_rat_bool,P3: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P3)))
    <=> ( ( pp(P3)
         => pp(aa_rat_bool(P,one_one_rat)) )
        & ( ~ pp(P3)
         => pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool
tff(fact_7565_split__of__bool,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> ( ( pp(P3)
         => pp(aa_int_bool(P,one_one_int)) )
        & ( ~ pp(P3)
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_7566_split__of__bool,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,aa_bool_nat(zero_n2687167440665602831ol_nat,P3)))
    <=> ( ( pp(P3)
         => pp(aa_nat_bool(P,one_one_nat)) )
        & ( ~ pp(P3)
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_7567_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = one_one_complex ) )
      & ( ~ pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_7568_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n3304061248610475627l_real(P3) = one_one_real ) )
      & ( ~ pp(P3)
       => ( zero_n3304061248610475627l_real(P3) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_7569_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n2052037380579107095ol_rat(P3) = one_one_rat ) )
      & ( ~ pp(P3)
       => ( zero_n2052037380579107095ol_rat(P3) = zero_zero_rat ) ) ) ).

% of_bool_def
tff(fact_7570_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = one_one_int ) )
      & ( ~ pp(P3)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_7571_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( aa_bool_nat(zero_n2687167440665602831ol_nat,P3) = one_one_nat ) )
      & ( ~ pp(P3)
       => ( aa_bool_nat(zero_n2687167440665602831ol_nat,P3) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_7572_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_7573_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_7574_bits__induct,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(divide_divide_int(A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => pp(aa_int_bool(P,A3)) )
     => ( ! [A3: int,B3: bool] :
            ( pp(aa_int_bool(P,A3))
           => ( ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_int_bool(P,A)) ) ) ).

% bits_induct
tff(fact_7575_bits__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => pp(aa_nat_bool(P,A3)) )
     => ( ! [A3: nat,B3: bool] :
            ( pp(aa_nat_bool(P,A3))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_nat_bool(P,A)) ) ) ).

% bits_induct
tff(fact_7576_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_7577_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_7578_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_7579_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : modulo8411746178871703098atural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M)) ).

% exp_mod_exp
tff(fact_7580_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(modulo_modulo_int(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))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_7581_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_7582_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(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(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_7583_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_7584_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_7585_arsinh__real__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_7586_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_7587_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_7588_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(divide_divide_int(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))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),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,M),N))) ).

% exp_div_exp_eq
tff(fact_7589_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),zero_zero_nat))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_7590_cos__arctan,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) = 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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7591_sin__arctan,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(arctan,X)) = aa_real_real(divide_divide_real(X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7592_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7593_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_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(X),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_7594_sin__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_7595_cos__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_7596_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_7597_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_7598_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(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_7599_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_7600_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),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(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_7601_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_7602_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_7603_mask__numeral,axiom,
    ! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_7604_mask__numeral,axiom,
    ! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_7605_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_7606_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A3: bool,B3: bool] : A1 = vEBT_Leaf(A3,B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M2))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                     => ( ( M2 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                         => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M2))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                       => ( ( M2 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                           => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                   => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Deg2,TreeList,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M2))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                         => ( ( M2 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                             => ( ! [I4: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
                                     => ~ ( ( Mi != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I4 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Deg2,TreeList,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M2))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                           => ( ( M2 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                               => ( ! [I4: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                                         => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
                                       => ~ ( ( Mi != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I4 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_7607_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_7608_mi__ma__2__deg,axiom,
    ! [Mi2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),Deg,TreeList2,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7609_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_7610_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_7611_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_7612_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_7613_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_7614_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_7615_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_7616_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_7617_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,N) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_7618_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_7619_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_7620_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),Deg,TreeList2,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X = Mi2 )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_7621_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_7622_cis__inverse,axiom,
    ! [A: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(cis,A)) = aa_real_complex(cis,aa_real_real(uminus_uminus_real,A)) ).

% cis_inverse
tff(fact_7623_member__inv,axiom,
    ! [Mi2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),Deg,TreeList2,Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
        & ( ( X = Mi2 )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(TreeList2)))
            & pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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_7624_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi2: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_7625_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_7626_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_7627_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ).

% take_bit_minus_one_eq_mask
tff(fact_7628_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% take_bit_minus_one_eq_mask
tff(fact_7629_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_7630_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_7631_of__int__mask__eq,axiom,
    ! [N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ).

% of_int_mask_eq
tff(fact_7632_of__int__mask__eq,axiom,
    ! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% of_int_mask_eq
tff(fact_7633_of__nat__mask__eq,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ).

% of_nat_mask_eq
tff(fact_7634_of__nat__mask__eq,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% of_nat_mask_eq
tff(fact_7635_of__nat__mask__eq,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% of_nat_mask_eq
tff(fact_7636_nat__mask__eq,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% nat_mask_eq
tff(fact_7637_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M2: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M2) ).

% prod_decode_aux.cases
tff(fact_7638_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_set_nat(bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_7639_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> ord_less_set_int(bot_bot_set_int,A) ) ).

% bot.not_eq_extremum
tff(fact_7640_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_7641_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_7642_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ ord_less_set_int(A,bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_7643_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_7644_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi2: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),zero_zero_nat,Va2,Vb)),X))
    <=> ( ( X = Mi2 )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_7645_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> ord_less_eq_set_int(X,Y) ) ).

% diff_shunt_var
tff(fact_7646_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% diff_shunt_var
tff(fact_7647_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ord_less_eq_set_int(A2,uminus1532241313380277803et_int(A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_7648_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(A2)))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_7649_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_nonnegative_int
tff(fact_7650_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_7651_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_7652_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_7653_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_7654_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_7655_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_7656_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_7657_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_7658_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_7659_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_7660_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_7661_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_7662_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_7663_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_7664_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7665_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7666_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_7667_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% mask_half_int
tff(fact_7668_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% mask_int_def
tff(fact_7669_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_7670_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_7671_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( 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))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi2 = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
                     => ( ( ( Mi2 != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_7672_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( 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))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi2 = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
                     => ( ( ( Mi2 != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_7673_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_7674_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_7675_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
    <=> ( ( ? [A4: bool,B4: bool] : A1 = vEBT_Leaf(A4,B4)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( 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,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: 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) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N4)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(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 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N4) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: 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) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( 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,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),aa_nat_nat(suc,N4))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(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 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),aa_nat_nat(suc,N4))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N4) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_7676_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q4)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)) = 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))),Q4)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_7677_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) = 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))),Q4)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) = 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))),Q4)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_7678_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q4: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q4)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R2),aa_num_Code_integer(numera6620942414471956472nteger,L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q4),R2) ) ) ) ).

% divmod_step_eq
tff(fact_7679_card_Oempty,axiom,
    finite_card_complex(bot_bot_set_complex) = zero_zero_nat ).

% card.empty
tff(fact_7680_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_7681_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_7682_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_7683_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_7684_divides__aux__eq,axiom,
    ! [Q4: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q4,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_7685_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_7686_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( unique6319869463603278526ux_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_7687_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,Ys2: list_Code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys2))))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys2),N) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s3445333598471063425nteger(Ys2))),aa_nat_Code_integer(nth_Code_integer(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s3445333598471063425nteger(Ys2)))) ) ) ).

% product_nth
tff(fact_7688_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr136253548056570280r_bool(produc2469108709554126195r_bool(Xs,Ys2),N) = produc5135875973608410785r_bool(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_7689_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_7690_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys2),N) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_7691_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),aa_list_nat_nat(size_size_list_nat,Ys2))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_7692_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys2))))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys2),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_7693_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys2),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_7694_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys2),N) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_7695_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),aa_list_nat_nat(size_size_list_nat,Ys2))))
     => ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys2),N) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_7696_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys2))))
     => ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys2),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_7697_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_7698_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_7699_complex__i__mult__minus,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),X)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% complex_i_mult_minus
tff(fact_7700_inverse__i,axiom,
    aa_complex_complex(invers8013647133539491842omplex,imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% inverse_i
tff(fact_7701_divide__i,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).

% divide_i
tff(fact_7702_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_7703_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_7704_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_7705_divide__numeral__i,axiom,
    ! [Z: complex,N: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(N)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(N)) ).

% divide_numeral_i
tff(fact_7706_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_7707_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_7708_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_7709_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_7710_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_7711_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_7712_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,D4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_7713_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_7714_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_7715_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_7716_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_complex(complex2(X),Y) = imaginary_unit )
    <=> ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_7717_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_7718_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_7719_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_7720_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_7721_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X3)
       => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),X3)
         => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_7722_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2),X3)
           => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_7723_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_7724_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_7725_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7726_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7727_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7728_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7729_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7730_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7731_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( 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(Q4),R2))
       => 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(Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_7732_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_7733_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_7734_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_7735_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,M)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_7736_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_7737_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_7738_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_7739_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_7740_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_7741_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_7742_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_7743_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K)) ).

% eucl_rel_int_by0
tff(fact_7744_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
       => eucl_rel_int(K,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_7745_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( eucl_rel_int(A,B,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),aa_int_int(uminus_uminus_int,Q4)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),one_one_int))),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_7746_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ).

% Arg_bounded
tff(fact_7747_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_7748_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L)),R2) )
         => eucl_rel_int(K,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_7749_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_7750_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ) ).

% Arg_correct
tff(fact_7751_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( 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,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L3) )
            & pp(aa_int_bool(aa_int_fun_int_bool(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,Q6),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_7752_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q2: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R4) )
               => ( ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(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,Q2),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_7753_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_7754_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_7755_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_7756_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( eucl_rel_int(A,B,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
       => 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(Q4),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_7757_one__div__minus__numeral,axiom,
    ! [N: num] : aa_int_int(divide_divide_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_7758_minus__one__div__numeral,axiom,
    ! [N: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_7759_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_7760_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N))) ).

% numeral_div_minus_numeral
tff(fact_7761_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_div_numeral
tff(fact_7762_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_7763_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_7764_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X2: num] : size_option_num(X,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_7765_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),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,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_7766_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = zero_zero_int ) )
      & ( ~ ( ( K = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = L ) )
          & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = 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(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_7767_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_7768_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_7769_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_7770_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_7771_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_7772_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_7773_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_7774_bit__0__eq,axiom,
    bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_bool ).

% bit_0_eq
tff(fact_7775_bit__0__eq,axiom,
    aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_bool ).

% bit_0_eq
tff(fact_7776_sinh__0,axiom,
    aa_real_real(sinh_real,zero_zero_real) = zero_zero_real ).

% sinh_0
tff(fact_7777_sinh__minus,axiom,
    ! [X: real] : aa_real_real(sinh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sinh_real,X)) ).

% sinh_minus
tff(fact_7778_sinh__minus,axiom,
    ! [X: complex] : aa_complex_complex(sinh_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sinh_complex,X)) ).

% sinh_minus
tff(fact_7779_bit_Oconj__one__right,axiom,
    ! [X: code_integer] : bit_se3949692690581998587nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ).

% bit.conj_one_right
tff(fact_7780_bit_Oconj__one__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).

% bit.conj_one_right
tff(fact_7781_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_7782_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_7783_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_7784_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_7785_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_7786_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_7787_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7788_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7789_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7790_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7791_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,M)),N)) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_7792_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_7793_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ).

% and_numerals(2)
tff(fact_7794_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_7795_and__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_7796_and__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_7797_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,M)),N)) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_7798_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_7799_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_7800_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_7801_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_7802_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_7803_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_7804_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_7805_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_7806_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_7807_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_7808_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_7809_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_7810_bit__0,axiom,
    ! [A: int] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),zero_zero_nat))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% bit_0
tff(fact_7811_bit__0,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),zero_zero_nat))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).

% bit_0
tff(fact_7812_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_7813_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_7814_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(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,N)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_7815_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(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,N)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_7816_bit__mod__2__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_nat_bool(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),N))
    <=> ( ( N = zero_zero_nat )
        & ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) ) ) ).

% bit_mod_2_iff
tff(fact_7817_bit__mod__2__iff,axiom,
    ! [A: code_natural,N: nat] :
      ( pp(aa_nat_bool(bit_se8040316288895769887atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))),N))
    <=> ( ( N = zero_zero_nat )
        & ~ dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A) ) ) ).

% bit_mod_2_iff
tff(fact_7818_bit__mod__2__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),N))
    <=> ( ( N = zero_zero_nat )
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).

% bit_mod_2_iff
tff(fact_7819_bit__mod__2__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
    <=> ( ( N = zero_zero_nat )
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ) ).

% bit_mod_2_iff
tff(fact_7820_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_7821_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_7822_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_se725231765392027082nd_int(K),L)) = bit_se3949692690581998587nteger(aa_int_Code_integer(ring_18347121197199848620nteger,K),aa_int_Code_integer(ring_18347121197199848620nteger,L)) ).

% of_int_and_eq
tff(fact_7823_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se725231765392027082nd_int(K),L)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_and_eq
tff(fact_7824_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(bit_se9216721137139052372nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N)) ) ).

% bit_of_nat_iff_bit
tff(fact_7825_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_nat_int(semiri1314217659103216013at_int,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N)) ) ).

% bit_of_nat_iff_bit
tff(fact_7826_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N)) ) ).

% bit_of_nat_iff_bit
tff(fact_7827_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = bit_se3949692690581998587nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_and_eq
tff(fact_7828_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_and_eq
tff(fact_7829_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_and_eq
tff(fact_7830_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N2))
          | ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(B),N2)) )
     => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N))
      <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N))
          | pp(aa_nat_bool(bit_se1146084159140164899it_int(B),N)) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_7831_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N2))
          | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,B),N2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,B),N)) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_7832_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7833_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7834_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,N))) ).

% not_bit_1_Suc
tff(fact_7835_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,N))) ).

% not_bit_1_Suc
tff(fact_7836_bit__1__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(one_one_int),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_7837_bit__1__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,one_one_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_7838_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(one_one_int),aa_num_nat(numeral_numeral_nat,N))) ).

% bit_numeral_simps(1)
tff(fact_7839_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% bit_numeral_simps(1)
tff(fact_7840_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(M),A)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N)) ) ) ).

% bit_take_bit_iff
tff(fact_7841_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N)) ) ) ).

% bit_take_bit_iff
tff(fact_7842_bit__of__bool__iff,axiom,
    ! [B: bool,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_bool_int(zero_n2684676970156552555ol_int,B)),N))
    <=> ( pp(B)
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_7843_bit__of__bool__iff,axiom,
    ! [B: bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B)),N))
    <=> ( pp(B)
        & ( N = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_7844_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(X),Y))) ) ).

% AND_lower
tff(fact_7845_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),X)) ) ).

% AND_upper1
tff(fact_7846_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Y)) ) ).

% AND_upper2
tff(fact_7847_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_7848_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_7849_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = zero_zero_int )
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N)) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7850_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_zero_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N)) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7851_inc__BitM__eq,axiom,
    ! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_7852_BitM__inc__eq,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_7853_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K)) ) ).

% and_less_eq
tff(fact_7854_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_7855_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_7856_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% bit_not_int_iff'
tff(fact_7857_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_7858_sinh__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,X)),aa_real_real(sinh_real,Y))) ).

% sinh_add
tff(fact_7859_cosh__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y))) ).

% cosh_add
tff(fact_7860_sinh__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(sinh_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,X)),aa_real_real(sinh_real,Y))) ).

% sinh_diff
tff(fact_7861_cosh__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cosh_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y))) ).

% cosh_diff
tff(fact_7862_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_7863_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_7864_cosh__plus__sinh,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cosh_complex,X)),aa_complex_complex(sinh_complex,X)) = aa_complex_complex(exp_complex,X) ).

% cosh_plus_sinh
tff(fact_7865_cosh__plus__sinh,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cosh_real,X)),aa_real_real(sinh_real,X)) = aa_real_real(exp_real,X) ).

% cosh_plus_sinh
tff(fact_7866_sinh__plus__cosh,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,X)) = aa_complex_complex(exp_complex,X) ).

% sinh_plus_cosh
tff(fact_7867_sinh__plus__cosh,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) = aa_real_real(exp_real,X) ).

% sinh_plus_cosh
tff(fact_7868_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_7869_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(M,K),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
          & pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_7870_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(bit_concat_bit(N,K),aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_7871_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
     => ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N)) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7872_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N)) ) ).

% exp_eq_0_imp_not_bit
tff(fact_7873_bit__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),N)) ) ).

% bit_Suc
tff(fact_7874_bit__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N)) ) ).

% bit_Suc
tff(fact_7875_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7876_one__and__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(one_one_Code_natural,A) = modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7877_one__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7878_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7879_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7880_and__one__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(A,one_one_Code_natural) = modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7881_and__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7882_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7883_numeral__BitM,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bitM,N)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,N))),one_one_complex) ).

% numeral_BitM
tff(fact_7884_numeral__BitM,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,N)) = aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,N))),one_one_real) ).

% numeral_BitM
tff(fact_7885_numeral__BitM,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,N)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N))),one_one_rat) ).

% numeral_BitM
tff(fact_7886_numeral__BitM,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),one_one_int) ).

% numeral_BitM
tff(fact_7887_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
             => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),M5))
              <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_7888_sinh__minus__cosh,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) ).

% sinh_minus_cosh
tff(fact_7889_sinh__minus__cosh,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) ).

% sinh_minus_cosh
tff(fact_7890_cosh__minus__sinh,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_real_real(cosh_real,X)),aa_real_real(sinh_real,X)) = aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)) ).

% cosh_minus_sinh
tff(fact_7891_cosh__minus__sinh,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cosh_complex,X)),aa_complex_complex(sinh_complex,X)) = aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)) ).

% cosh_minus_sinh
tff(fact_7892_even__bit__succ__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),N))
      <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7893_even__bit__succ__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),N))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7894_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N: nat] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N))
      <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7895_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),N))
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_7896_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [J2: nat] : ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2)))
     => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_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))),N))
      <=> ( ( ( N = zero_zero_nat )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) )
          & ( ( N != zero_zero_nat )
           => pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),N)) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7897_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [J2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),N))
      <=> ( ( ( N = zero_zero_nat )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) )
          & ( ( N != zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)),N)) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7898_bit__rec,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N))
    <=> ( ( ( N = zero_zero_nat )
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) )
        & ( ( N != zero_zero_nat )
         => pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ) ).

% bit_rec
tff(fact_7899_bit__rec,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
    <=> ( ( ( N = zero_zero_nat )
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) )
        & ( ( N != zero_zero_nat )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ) ).

% bit_rec
tff(fact_7900_and__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_7901_sinh__field__def,axiom,
    ! [Z: real] : aa_real_real(sinh_real,Z) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_7902_sinh__field__def,axiom,
    ! [Z: complex] : aa_complex_complex(sinh_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Z)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_7903_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_7904_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_7905_cosh__square__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% cosh_square_eq
tff(fact_7906_cosh__square__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% cosh_square_eq
tff(fact_7907_hyperbolic__pythagoras,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% hyperbolic_pythagoras
tff(fact_7908_hyperbolic__pythagoras,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% hyperbolic_pythagoras
tff(fact_7909_sinh__square__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% sinh_square_eq
tff(fact_7910_sinh__square__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% sinh_square_eq
tff(fact_7911_set__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),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_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% set_bit_eq
tff(fact_7912_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),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_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% unset_bit_eq
tff(fact_7913_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),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))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_Suc_from_most
tff(fact_7914_cosh__double,axiom,
    ! [X: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_7915_cosh__double,axiom,
    ! [X: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_7916_and__int_Oelims,axiom,
    ! [X: int,Xa3: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa3) = Y )
     => ( ( ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xa3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
         => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa3))))) ) )
        & ( ~ ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
              & member_int(Xa3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
         => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa3))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_7917_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
          & 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(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
      & ( ~ ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & 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(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_7918_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),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,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_7919_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_7920_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_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_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_7921_bit_Oxor__self,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),X) = zero_zero_int ).

% bit.xor_self
tff(fact_7922_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_7923_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_7924_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_7925_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_7926_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_7927_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_7928_subset__Compl__singleton,axiom,
    ! [A2: set_complex,B: complex] :
      ( ord_le211207098394363844omplex(A2,uminus8566677241136511917omplex(insert_complex(B,bot_bot_set_complex)))
    <=> ~ member_complex(B,A2) ) ).

% subset_Compl_singleton
tff(fact_7929_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ord_less_eq_set_real(A2,uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ member_real(B,A2) ) ).

% subset_Compl_singleton
tff(fact_7930_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( ord_le6893508408891458716et_nat(A2,uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ member_set_nat(B,A2) ) ).

% subset_Compl_singleton
tff(fact_7931_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ord_less_eq_set_int(A2,uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ member_int(B,A2) ) ).

% subset_Compl_singleton
tff(fact_7932_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_7933_card__Diff__insert,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( member_real(A,A2)
     => ( ~ member_real(A,B2)
       => ( finite_card_real(minus_minus_set_real(A2,insert_real(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(minus_minus_set_real(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7934_card__Diff__insert,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( member_complex(A,A2)
     => ( ~ member_complex(A,B2)
       => ( finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(minus_811609699411566653omplex(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7935_card__Diff__insert,axiom,
    ! [A: list_nat,A2: set_list_nat,B2: set_list_nat] :
      ( member_list_nat(A,A2)
     => ( ~ member_list_nat(A,B2)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7936_card__Diff__insert,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( member_set_nat(A,A2)
     => ( ~ member_set_nat(A,B2)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7937_card__Diff__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( member_int(A,A2)
     => ( ~ member_int(A,B2)
       => ( finite_card_int(minus_minus_set_int(A2,insert_int(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(minus_minus_set_int(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7938_card__Diff__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(A),B2))
       => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,insert_nat(A,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,B2))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_7939_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_numerals(1)
tff(fact_7940_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% xor_numerals(1)
tff(fact_7941_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_numerals(2)
tff(fact_7942_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y)) ).

% xor_numerals(2)
tff(fact_7943_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_numerals(5)
tff(fact_7944_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X)) ).

% xor_numerals(5)
tff(fact_7945_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_numerals(8)
tff(fact_7946_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,X)) ).

% xor_numerals(8)
tff(fact_7947_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_7948_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7949_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_7950_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_7951_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_7952_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_7953_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7954_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_7955_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_7956_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_7957_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_7958_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(6)
tff(fact_7959_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_7960_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(4)
tff(fact_7961_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N)) = bit_se3222712562003087583nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_xor_eq
tff(fact_7962_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_xor_eq
tff(fact_7963_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N)) = aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_xor_eq
tff(fact_7964_of__int__xor__eq,axiom,
    ! [K: int,L: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_se6526347334894502574or_int(K),L)) = bit_se3222712562003087583nteger(aa_int_Code_integer(ring_18347121197199848620nteger,K),aa_int_Code_integer(ring_18347121197199848620nteger,L)) ).

% of_int_xor_eq
tff(fact_7965_of__int__xor__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se6526347334894502574or_int(K),L)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_xor_eq
tff(fact_7966_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_7967_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_7968_Compl__insert,axiom,
    ! [X: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(X,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_7969_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_7970_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_7971_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% and_nat_def
tff(fact_7972_card__1__singletonE,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = one_one_nat )
     => ~ ! [X3: complex] : A2 != insert_complex(X3,bot_bot_set_complex) ) ).

% card_1_singletonE
tff(fact_7973_card__1__singletonE,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = one_one_nat )
     => ~ ! [X3: list_nat] : A2 != insert_list_nat(X3,bot_bot_set_list_nat) ) ).

% card_1_singletonE
tff(fact_7974_card__1__singletonE,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = one_one_nat )
     => ~ ! [X3: set_nat] : A2 != insert_set_nat(X3,bot_bot_set_set_nat) ) ).

% card_1_singletonE
tff(fact_7975_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = one_one_nat )
     => ~ ! [X3: nat] : A2 != insert_nat(X3,bot_bot_set_nat) ) ).

% card_1_singletonE
tff(fact_7976_card__1__singletonE,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = one_one_nat )
     => ~ ! [X3: int] : A2 != insert_int(X3,bot_bot_set_int) ) ).

% card_1_singletonE
tff(fact_7977_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ) ).

% bit_nat_iff
tff(fact_7978_card__1__singleton__iff,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: complex] : A2 = insert_complex(X5,bot_bot_set_complex) ) ).

% card_1_singleton_iff
tff(fact_7979_card__1__singleton__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: list_nat] : A2 = insert_list_nat(X5,bot_bot_set_list_nat) ) ).

% card_1_singleton_iff
tff(fact_7980_card__1__singleton__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: set_nat] : A2 = insert_set_nat(X5,bot_bot_set_set_nat) ) ).

% card_1_singleton_iff
tff(fact_7981_card__1__singleton__iff,axiom,
    ! [A2: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: nat] : A2 = insert_nat(X5,bot_bot_set_nat) ) ).

% card_1_singleton_iff
tff(fact_7982_card__1__singleton__iff,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: int] : A2 = insert_int(X5,bot_bot_set_int) ) ).

% card_1_singleton_iff
tff(fact_7983_card__eq__SucD,axiom,
    ! [A2: set_real,K: nat] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,K) )
     => ? [B3: real,B7: set_real] :
          ( ( A2 = insert_real(B3,B7) )
          & ~ member_real(B3,B7)
          & ( finite_card_real(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
tff(fact_7984_card__eq__SucD,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,K) )
     => ? [B3: complex,B7: set_complex] :
          ( ( A2 = insert_complex(B3,B7) )
          & ~ member_complex(B3,B7)
          & ( finite_card_complex(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
tff(fact_7985_card__eq__SucD,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,K) )
     => ? [B3: list_nat,B7: set_list_nat] :
          ( ( A2 = insert_list_nat(B3,B7) )
          & ~ member_list_nat(B3,B7)
          & ( finite_card_list_nat(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_7986_card__eq__SucD,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,K) )
     => ? [B3: set_nat,B7: set_set_nat] :
          ( ( A2 = insert_set_nat(B3,B7) )
          & ~ member_set_nat(B3,B7)
          & ( finite_card_set_nat(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_7987_card__eq__SucD,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = aa_nat_nat(suc,K) )
     => ? [B3: nat,B7: set_nat] :
          ( ( A2 = insert_nat(B3,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(B3),B7))
          & ( aa_set_nat_nat(finite_card_nat,B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_7988_card__eq__SucD,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,K) )
     => ? [B3: int,B7: set_int] :
          ( ( A2 = insert_int(B3,B7) )
          & ~ member_int(B3,B7)
          & ( finite_card_int(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
tff(fact_7989_card__Suc__eq,axiom,
    ! [A2: set_real,K: nat] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,K) )
    <=> ? [B4: real,B8: set_real] :
          ( ( A2 = insert_real(B4,B8) )
          & ~ member_real(B4,B8)
          & ( finite_card_real(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_Suc_eq
tff(fact_7990_card__Suc__eq,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,K) )
    <=> ? [B4: complex,B8: set_complex] :
          ( ( A2 = insert_complex(B4,B8) )
          & ~ member_complex(B4,B8)
          & ( finite_card_complex(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_Suc_eq
tff(fact_7991_card__Suc__eq,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B4: list_nat,B8: set_list_nat] :
          ( ( A2 = insert_list_nat(B4,B8) )
          & ~ member_list_nat(B4,B8)
          & ( finite_card_list_nat(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_7992_card__Suc__eq,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B4: set_nat,B8: set_set_nat] :
          ( ( A2 = insert_set_nat(B4,B8) )
          & ~ member_set_nat(B4,B8)
          & ( finite_card_set_nat(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_7993_card__Suc__eq,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = aa_nat_nat(suc,K) )
    <=> ? [B4: nat,B8: set_nat] :
          ( ( A2 = insert_nat(B4,B8) )
          & ~ pp(aa_set_nat_bool(member_nat(B4),B8))
          & ( aa_set_nat_nat(finite_card_nat,B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_7994_card__Suc__eq,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,K) )
    <=> ? [B4: int,B8: set_int] :
          ( ( A2 = insert_int(B4,B8) )
          & ~ member_int(B4,B8)
          & ( finite_card_int(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_Suc_eq
tff(fact_7995_card__Diff__singleton__if,axiom,
    ! [X: real,A2: set_real] :
      ( ( member_real(X,A2)
       => ( finite_card_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A2)),one_one_nat) ) )
      & ( ~ member_real(X,A2)
       => ( finite_card_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))) = finite_card_real(A2) ) ) ) ).

% card_Diff_singleton_if
tff(fact_7996_card__Diff__singleton__if,axiom,
    ! [X: complex,A2: set_complex] :
      ( ( member_complex(X,A2)
       => ( finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),one_one_nat) ) )
      & ( ~ member_complex(X,A2)
       => ( finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex))) = finite_card_complex(A2) ) ) ) ).

% card_Diff_singleton_if
tff(fact_7997_card__Diff__singleton__if,axiom,
    ! [X: list_nat,A2: set_list_nat] :
      ( ( member_list_nat(X,A2)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(X,bot_bot_set_list_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),one_one_nat) ) )
      & ( ~ member_list_nat(X,A2)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(X,bot_bot_set_list_nat))) = finite_card_list_nat(A2) ) ) ) ).

% card_Diff_singleton_if
tff(fact_7998_card__Diff__singleton__if,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ( member_set_nat(X,A2)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),one_one_nat) ) )
      & ( ~ member_set_nat(X,A2)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))) = finite_card_set_nat(A2) ) ) ) ).

% card_Diff_singleton_if
tff(fact_7999_card__Diff__singleton__if,axiom,
    ! [X: int,A2: set_int] :
      ( ( member_int(X,A2)
       => ( finite_card_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),one_one_nat) ) )
      & ( ~ member_int(X,A2)
       => ( finite_card_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))) = finite_card_int(A2) ) ) ) ).

% card_Diff_singleton_if
tff(fact_8000_card__Diff__singleton__if,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X),A2))
       => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A2)),one_one_nat) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))) = aa_set_nat_nat(finite_card_nat,A2) ) ) ) ).

% card_Diff_singleton_if
tff(fact_8001_card__Diff__singleton,axiom,
    ! [X: real,A2: set_real] :
      ( member_real(X,A2)
     => ( finite_card_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_8002_card__Diff__singleton,axiom,
    ! [X: complex,A2: set_complex] :
      ( member_complex(X,A2)
     => ( finite_card_complex(minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_8003_card__Diff__singleton,axiom,
    ! [X: list_nat,A2: set_list_nat] :
      ( member_list_nat(X,A2)
     => ( finite_card_list_nat(minus_7954133019191499631st_nat(A2,insert_list_nat(X,bot_bot_set_list_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_8004_card__Diff__singleton,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( member_set_nat(X,A2)
     => ( finite_card_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_8005_card__Diff__singleton,axiom,
    ! [X: int,A2: set_int] :
      ( member_int(X,A2)
     => ( finite_card_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_8006_card__Diff__singleton,axiom,
    ! [X: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),A2))
     => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_8007_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8008_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = 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_8009_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(insert_complex(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_8010_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_list_nat,X: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(insert_list_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_8011_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(insert_set_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_8012_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,insert_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_8013_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(insert_int(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_8014_sinh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sinh_real,X) = zero_zero_real )
    <=> member_real(aa_real_real(exp_real,X),insert_real(one_one_real,insert_real(aa_real_real(uminus_uminus_real,one_one_real),bot_bot_set_real))) ) ).

% sinh_zero_iff
tff(fact_8015_sinh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sinh_complex,X) = zero_zero_complex )
    <=> member_complex(aa_complex_complex(exp_complex,X),insert_complex(one_one_complex,insert_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex),bot_bot_set_complex))) ) ).

% sinh_zero_iff
tff(fact_8016_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(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,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(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_8017_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_8018_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = 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,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_8019_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_8020_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))) ).

% xor_one_eq
tff(fact_8021_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))) ).

% xor_one_eq
tff(fact_8022_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))) ).

% one_xor_eq
tff(fact_8023_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))) ).

% one_xor_eq
tff(fact_8024_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,groups4051137065250181068ol_nat(zero_n2687167440665602831ol_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)),Bs)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Bs)))
        & pp(aa_nat_bool(nth_bool(Bs),N)) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_8025_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_bool,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Bs)))
        & pp(aa_nat_bool(nth_bool(Bs),N)) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_8026_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(K),L))
     => ( ( ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & 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(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
              & 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(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_8027_and__int_Opelims,axiom,
    ! [X: int,Xa3: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa3) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa3))
       => ~ ( ( ( ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xa3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa3))))) ) )
              & ( ~ ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(Xa3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa3))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa3)) ) ) ) ).

% and_int.pelims
tff(fact_8028_push__bit__numeral__minus__1,axiom,
    ! [N: num] : bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,N))) ).

% push_bit_numeral_minus_1
tff(fact_8029_push__bit__numeral__minus__1,axiom,
    ! [N: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,N)),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))),aa_num_nat(numeral_numeral_nat,N))) ).

% push_bit_numeral_minus_1
tff(fact_8030_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa3))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa3 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa3 != zero_zero_nat )
               => ( ( ( Xa3 = one_one_nat )
                   => pp(B3) )
                  & ( Xa3 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
           => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)
                   => ( ( Xa3 != Mi )
                     => ( ( Xa3 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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(vEBT_VEBT_low,Xa3),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)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_8031_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa3))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa3 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => pp(B3) )
                      & ( Xa3 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa3 != Mi )
                         => ( ( Xa3 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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(vEBT_VEBT_low,Xa3),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)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_8032_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_8033_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_8034_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_8035_push__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),zero_zero_int) = zero_zero_int ).

% push_bit_of_0
tff(fact_8036_push__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),zero_zero_nat) = zero_zero_nat ).

% push_bit_of_0
tff(fact_8037_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(bit_se545348938243370406it_int(N),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
tff(fact_8038_push__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
tff(fact_8039_push__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(M),aa_int_int(bit_se545348938243370406it_int(N),A)) = aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),A) ).

% push_bit_push_bit
tff(fact_8040_push__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),A) ).

% push_bit_push_bit
tff(fact_8041_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa_int_int(bit_concat_bit(N,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(N),L) ).

% concat_bit_of_zero_1
tff(fact_8042_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N))) = N ).

% card_Collect_less_nat
tff(fact_8043_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_8044_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_8045_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ag,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_8046_push__bit__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,K)) = aa_int_int(bit_se545348938243370406it_int(N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) ).

% push_bit_Suc_numeral
tff(fact_8047_push__bit__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% push_bit_Suc_numeral
tff(fact_8048_push__bit__Suc__minus__numeral,axiom,
    ! [N: nat,K: num] : bit_se7788150548672797655nteger(aa_nat_nat(suc,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = bit_se7788150548672797655nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_8049_push__bit__Suc__minus__numeral,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_8050_push__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,N)),A) = aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% push_bit_Suc
tff(fact_8051_push__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% push_bit_Suc
tff(fact_8052_push__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),one_one_int) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ).

% push_bit_of_1
tff(fact_8053_push__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_1
tff(fact_8054_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),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))),N) ).

% push_bit_of_Suc_0
tff(fact_8055_even__push__bit__iff,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se545348938243370406it_int(N),A)))
    <=> ( ( N != zero_zero_nat )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).

% even_push_bit_iff
tff(fact_8056_even__push__bit__iff,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)))
    <=> ( ( N != zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ) ).

% even_push_bit_iff
tff(fact_8057_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,L),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = bit_se7788150548672797655nteger(aa_num_nat(pred_numeral,L),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K)))) ).

% push_bit_minus_numeral
tff(fact_8058_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_se545348938243370406it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ).

% push_bit_minus_numeral
tff(fact_8059_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(N),K) = aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% flip_bit_int_def
tff(fact_8060_lambda__zero,axiom,
    aTP_Lamm_ah = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_8061_lambda__zero,axiom,
    aTP_Lamm_ai = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_8062_lambda__zero,axiom,
    aTP_Lamm_aj = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_8063_lambda__zero,axiom,
    aTP_Lamm_ak = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_8064_Compl__eq,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(aTP_Lamm_al(A2)) ).

% Compl_eq
tff(fact_8065_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_am(A2)) ).

% Compl_eq
tff(fact_8066_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_an(A2)) ).

% Compl_eq
tff(fact_8067_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_ao(A2)) ).

% Compl_eq
tff(fact_8068_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_ap(A2)) ).

% Compl_eq
tff(fact_8069_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_aq(A2)) ).

% Compl_eq
tff(fact_8070_Collect__neg__eq,axiom,
    ! [P: fun_real_bool] : collect_real(aTP_Lamm_ar(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_8071_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_bool] : collect_list_nat(aTP_Lamm_as(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_8072_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_bool] : collect_set_nat(aTP_Lamm_at(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_8073_Collect__neg__eq,axiom,
    ! [P: fun_nat_bool] : collect_nat(aTP_Lamm_au(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_8074_Collect__neg__eq,axiom,
    ! [P: fun_int_bool] : collect_int(aTP_Lamm_av(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_8075_uminus__set__def,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(uminus8337675090112669241x_bool(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_8076_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus4638318004487240379l_bool(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_8077_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus5119773305880778631t_bool(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_8078_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus6226608070181861729t_bool(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_8079_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus1995910650241132439t_bool(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_8080_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus8094991720627284027t_bool(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_8081_push__bit__of__int,axiom,
    ! [N: nat,K: int] : bit_se7788150548672797655nteger(N,aa_int_Code_integer(ring_18347121197199848620nteger,K)) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_se545348938243370406it_int(N),K)) ).

% push_bit_of_int
tff(fact_8082_push__bit__of__int,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se545348938243370406it_int(N),K)) ).

% push_bit_of_int
tff(fact_8083_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] : bit_se7788150548672797655nteger(N,aa_nat_Code_integer(semiri4939895301339042750nteger,M)) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M)) ).

% push_bit_of_nat
tff(fact_8084_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] : aa_int_int(bit_se545348938243370406it_int(N),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M)) ).

% push_bit_of_nat
tff(fact_8085_push__bit__of__nat,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M)) ).

% push_bit_of_nat
tff(fact_8086_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),N)) = bit_se7788150548672797655nteger(M,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_push_bit
tff(fact_8087_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),N)) = aa_int_int(bit_se545348938243370406it_int(M),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_push_bit
tff(fact_8088_of__nat__push__bit,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_push_bit
tff(fact_8089_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(N),K)) ).

% push_bit_nat_eq
tff(fact_8090_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aw,A)),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aw,B)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8091_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,B))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8092_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ay,A)),collect_int(aa_int_fun_int_bool(aTP_Lamm_ay,B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8093_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aw,A)),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aw,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_8094_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ay,A)),collect_int(aa_int_fun_int_bool(aTP_Lamm_ay,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_8095_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_8096_push__bit__minus,axiom,
    ! [N: nat,A: code_integer] : bit_se7788150548672797655nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_se7788150548672797655nteger(N,A)) ).

% push_bit_minus
tff(fact_8097_push__bit__minus,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se545348938243370406it_int(N),A)) ).

% push_bit_minus
tff(fact_8098_lambda__one,axiom,
    aTP_Lamm_az = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_8099_lambda__one,axiom,
    aTP_Lamm_ba = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_8100_lambda__one,axiom,
    aTP_Lamm_bb = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_8101_lambda__one,axiom,
    aTP_Lamm_bc = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_8102_lambda__one,axiom,
    aTP_Lamm_bd = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_8103_push__bit__add,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(bit_se545348938243370406it_int(N),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(bit_se545348938243370406it_int(N),A)),aa_int_int(bit_se545348938243370406it_int(N),B)) ).

% push_bit_add
tff(fact_8104_push__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),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(bit_se547839408752420682it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),B)) ).

% push_bit_add
tff(fact_8105_numeral__code_I2_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_code(2)
tff(fact_8106_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_code(2)
tff(fact_8107_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_code(2)
tff(fact_8108_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_8109_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_8110_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_8111_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_8112_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_8113_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_code(3)
tff(fact_8114_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_code(3)
tff(fact_8115_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_8116_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_code(3)
tff(fact_8117_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_8118_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_8119_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_8120_card__less__Suc2,axiom,
    ! [M10: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M10))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_be(M10),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf(M10),I))) ) ) ).

% card_less_Suc2
tff(fact_8121_card__less__Suc,axiom,
    ! [M10: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M10))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_be(M10),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf(M10),I))) ) ) ).

% card_less_Suc
tff(fact_8122_card__less,axiom,
    ! [M10: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M10))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf(M10),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_8123_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_8124_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_8125_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bg(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_8126_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aTP_Lamm_bh(N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_8127_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(X),Y))) ) ) ).

% XOR_lower
tff(fact_8128_signed__take__bit__code,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(N,A) = if_Code_integer(aa_nat_bool(bit_se9216721137139052372nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A)),N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A)),bit_se7788150548672797655nteger(aa_nat_nat(suc,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))),bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A)) ).

% signed_take_bit_code
tff(fact_8129_signed__take__bit__code,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(if_int(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A)),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)))),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A)) ).

% signed_take_bit_code
tff(fact_8130_card__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(collect_real(aTP_Lamm_bi(N)))),N)) ) ).

% card_roots_unity
tff(fact_8131_card__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(collect_complex(aTP_Lamm_bh(N)))),N)) ) ).

% card_roots_unity
tff(fact_8132_push__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_int_int(bit_se545348938243370406it_int(M),A)) ).

% push_bit_take_bit
tff(fact_8133_push__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),A)) ).

% push_bit_take_bit
tff(fact_8134_take__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_se545348938243370406it_int(N),A)) = aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),A)) ).

% take_bit_push_bit
tff(fact_8135_take__bit__push__bit,axiom,
    ! [M: nat,N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),A)) ).

% take_bit_push_bit
tff(fact_8136_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_8137_diff__nat__eq__if,axiom,
    ! [Z3: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z3)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z3))) ) ) ) ).

% diff_nat_eq_if
tff(fact_8138_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_8139_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_8140_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_8141_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_8142_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_8143_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% xor_nat_def
tff(fact_8144_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bj,X)) ).

% set_decode_def
tff(fact_8145_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q4)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8146_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).

% concat_bit_eq
tff(fact_8147_flip__bit__eq__xor,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(N),A) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% flip_bit_eq_xor
tff(fact_8148_flip__bit__eq__xor,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).

% flip_bit_eq_xor
tff(fact_8149_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_bk(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_8150_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_bl(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_8151_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_bm(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_8152_pochhammer__code,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = set_fo1084959871951514735nteger(aTP_Lamm_bn(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_Code_integer) ) ) ) ).

% pochhammer_code
tff(fact_8153_pochhammer__code,axiom,
    ! [N: nat,A: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_bo(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_8154_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_bp(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_8155_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N))
    <=> ( aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) != zero_zero_int ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_8156_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,A),N))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) != zero_zero_nat ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_8157_gbinomial__code,axiom,
    ! [K: nat,A: complex] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = one_one_complex ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_bq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K)) ) ) ) ).

% gbinomial_code
tff(fact_8158_gbinomial__code,axiom,
    ! [K: nat,A: rat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = one_one_rat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_br(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ) ) ).

% gbinomial_code
tff(fact_8159_gbinomial__code,axiom,
    ! [K: nat,A: real] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = one_one_real ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_bs(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K)) ) ) ) ).

% gbinomial_code
tff(fact_8160_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2,S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),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)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_8161_push__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),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))),N)) ).

% push_bit_minus_one
tff(fact_8162_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2,Vd2)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),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)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_8163_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),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))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_8164_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi2: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),aa_nat_nat(suc,V),TreeList2,Vc)),X))
    <=> ( ( X = Mi2 )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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))))),size_s6755466524823107622T_VEBT(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),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)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_8165_vebt__member_Osimps_I5_J,axiom,
    ! [Mi2: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)),X))
    <=> ( ( X != Mi2 )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi2))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi2))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_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(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_8166_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_8167_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa3))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa3 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa3 != zero_zero_nat )
               => ( ( ( Xa3 = one_one_nat )
                   => pp(B3) )
                  & ( Xa3 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_8168_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa3))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa3 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa3 != zero_zero_nat )
                 => ( ( ( Xa3 = one_one_nat )
                     => pp(B3) )
                    & ( Xa3 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_8169_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa3))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa3 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => pp(B3) )
                      & ( Xa3 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_8170_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa3))
     => ( ! [Mi: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa3 = Mi )
                | ( Xa3 = Ma2 ) ) )
       => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2)
             => ~ ( ( Xa3 = Mi )
                  | ( Xa3 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
         => ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_8171_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(K2),L2))
           => ( ( ~ ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_8172_xor__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(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(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_8173_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa3))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa3 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa3 != zero_zero_nat )
                 => ( ( ( Xa3 = one_one_nat )
                     => pp(B3) )
                    & ( Xa3 = one_one_nat ) ) ) ) )
       => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)
             => ~ ( ( Xa3 != Mi )
                 => ( ( Xa3 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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(vEBT_VEBT_low,Xa3),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)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_8174_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa3))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa3 = Mi )
                      | ( Xa3 = Ma2 ) ) ) )
           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa3 = Mi )
                        | ( Xa3 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
             => ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_8175_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa3))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa3 = Mi )
                  | ( Xa3 = Ma2 ) ) )
           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2)
                 => ( ( Xa3 = Mi )
                    | ( Xa3 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
             => ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_8176_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_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(ring_1_of_int_int,aa_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(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(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8177_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_real_real(aa_rea7340605660300375871l_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(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_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(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8178_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8179_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8180_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8181_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aTP_Lamm_bt(X)) ) ).

% monoseq_arctan_series
tff(fact_8182_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups6464643781859351333omplex(aTP_Lamm_bu(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8183_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,N))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups129246275422532515t_real(aTP_Lamm_bv(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8184_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_bw(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8185_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aTP_Lamm_bx(X)) ) ) ) ).

% ln_series
tff(fact_8186_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aTP_Lamm_by(X)) ) ) ).

% arctan_series
tff(fact_8187_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_int(aa_complex_int(F,X3),ring_1_Ints_int) )
     => member_int(groups858564598930262913ex_int(F,A2),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_8188_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => member_int(aa_real_int(F,X3),ring_1_Ints_int) )
     => member_int(groups4694064378042380927al_int(F,A2),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_8189_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_real(aa_complex_real(F,X3),ring_1_Ints_real) )
     => member_real(groups766887009212190081x_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8190_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => member_real(aa_real_real(F,X3),ring_1_Ints_real) )
     => member_real(groups1681761925125756287l_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8191_Ints__prod,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => member_real(aa_nat_real(F,X3),ring_1_Ints_real) )
     => member_real(groups129246275422532515t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8192_Ints__prod,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => member_real(aa_int_real(F,X3),ring_1_Ints_real) )
     => member_real(groups2316167850115554303t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8193_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_complex(aa_complex_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups3708469109370488835omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8194_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => member_complex(aa_real_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups713298508707869441omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8195_Ints__prod,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => member_complex(aa_nat_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups6464643781859351333omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8196_Ints__prod,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => member_complex(aa_int_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups7440179247065528705omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8197_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_bz(F),A2) ).

% of_nat_prod
tff(fact_8198_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups129246275422532515t_real(aTP_Lamm_ca(F),A2) ).

% of_nat_prod
tff(fact_8199_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups3455450783089532116nteger(aTP_Lamm_cb(F),A2) ).

% of_nat_prod
tff(fact_8200_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups73079841787564623at_rat(aTP_Lamm_cc(F),A2) ).

% of_nat_prod
tff(fact_8201_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cd(F)),A2) ).

% of_nat_prod
tff(fact_8202_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_ce(F),A2) ).

% of_nat_prod
tff(fact_8203_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A2)) = groups129246275422532515t_real(aTP_Lamm_cf(F),A2) ).

% of_int_prod
tff(fact_8204_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_Code_integer(ring_18347121197199848620nteger,groups705719431365010083at_int(F,A2)) = groups3455450783089532116nteger(aTP_Lamm_cg(F),A2) ).

% of_int_prod
tff(fact_8205_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A2)) = groups73079841787564623at_rat(aTP_Lamm_ch(F),A2) ).

% of_int_prod
tff(fact_8206_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_complex(ring_17405671764205052669omplex,groups705719431365010083at_int(F,A2)) = groups6464643781859351333omplex(aTP_Lamm_ci(F),A2) ).

% of_int_prod
tff(fact_8207_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A2)) = groups705719431365010083at_int(aTP_Lamm_cj(F),A2) ).

% of_int_prod
tff(fact_8208_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_ck(F),A2) ).

% of_int_prod
tff(fact_8209_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_Code_integer(ring_18347121197199848620nteger,groups1705073143266064639nt_int(F,A2)) = groups3827104343326376752nteger(aTP_Lamm_cl(F),A2) ).

% of_int_prod
tff(fact_8210_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_cm(F),A2) ).

% of_int_prod
tff(fact_8211_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups1705073143266064639nt_int(F,A2)) = groups7440179247065528705omplex(aTP_Lamm_cn(F),A2) ).

% of_int_prod
tff(fact_8212_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_co(F),A2) ).

% of_int_prod
tff(fact_8213_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_cp(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_8214_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_cq(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_8215_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8216_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8217_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8218_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8219_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_8220_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cr(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8221_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = groups705719431365010083at_int(aTP_Lamm_cs(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8222_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ct(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8223_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_cu(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8224_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_cv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_8225_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_cw(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% prod.atLeastAtMost_rev
tff(fact_8226_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8227_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8228_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8229_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8230_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8231_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8232_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8233_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8234_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8235_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8236_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8237_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8238_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(aTP_Lamm_cx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8239_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_cy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8240_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cr(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8241_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_cs(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8242_fact__prod,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_8243_fact__prod,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_8244_fact__prod,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_8245_fact__prod,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_8246_fact__prod,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N))) ).

% fact_prod
tff(fact_8247_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_cz(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_8248_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups129246275422532515t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_da(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8249_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_db(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8250_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_dc(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8251_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_dd(F),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8252_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8253_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8254_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8255_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8256_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8257_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo6980174941875973593q_real(power_power_real(X)) ) ) ).

% monoseq_realpow
tff(fact_8258_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = groups129246275422532515t_real(aTP_Lamm_de(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_8259_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = groups3455450783089532116nteger(aTP_Lamm_df(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_8260_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aTP_Lamm_dg(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_8261_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_8262_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aTP_Lamm_di(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_8263_pochhammer__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),N) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_dj(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_8264_pochhammer__prod__rev,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_dk(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_8265_pochhammer__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_dl(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_8266_pochhammer__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dm(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_8267_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_dn(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_8268_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_8269_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups129246275422532515t_real(aTP_Lamm_do(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_8270_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_dp(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_8271_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dq(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_8272_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_dr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_8273_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_dj(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8274_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_dk(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8275_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_dl(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8276_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dm(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8277_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_dn(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_8278_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(aTP_Lamm_ds(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8279_gbinomial__Suc,axiom,
    ! [A: code_integer,K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),aa_nat_nat(suc,K)) = divide6298287555418463151nteger(groups3455450783089532116nteger(aTP_Lamm_dt(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8280_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_du(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8281_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(divide_divide_real(groups129246275422532515t_real(aTP_Lamm_dv(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8282_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dw,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8283_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(divide_divide_int(groups705719431365010083at_int(aTP_Lamm_dx(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8284_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_dy) ).

% pi_series
tff(fact_8285_suminf__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => ( suminf_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_8286_suminf__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_8287_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_8288_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8289_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8290_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_8291_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_8292_prod_Oempty,axiom,
    ! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).

% prod.empty
tff(fact_8293_prod_Oempty,axiom,
    ! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).

% prod.empty
tff(fact_8294_prod_Oempty,axiom,
    ! [G: fun_nat_nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),bot_bot_set_nat) = one_one_nat ).

% prod.empty
tff(fact_8295_prod_Oempty,axiom,
    ! [G: fun_nat_int] : groups705719431365010083at_int(G,bot_bot_set_nat) = one_one_int ).

% prod.empty
tff(fact_8296_prod_Oempty,axiom,
    ! [G: fun_int_int] : groups1705073143266064639nt_int(G,bot_bot_set_int) = one_one_int ).

% prod.empty
tff(fact_8297_prod__le__power,axiom,
    ! [A2: set_real,F: fun_real_real,N: real,K: nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A2)),K))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),aa_nat_real(power_power_real(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8298_prod__le__power,axiom,
    ! [A2: set_complex,F: fun_complex_real,N: real,K: nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A2)),K))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),aa_nat_real(power_power_real(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8299_prod__le__power,axiom,
    ! [A2: set_nat,F: fun_nat_real,N: real,K: nat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),K))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A2)),aa_nat_real(power_power_real(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8300_prod__le__power,axiom,
    ! [A2: set_int,F: fun_int_real,N: real,K: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A2)),K))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),aa_nat_real(power_power_real(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8301_prod__le__power,axiom,
    ! [A2: set_real,F: fun_real_rat,N: rat,K: nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A2)),K))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),aa_nat_rat(power_power_rat(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8302_prod__le__power,axiom,
    ! [A2: set_complex,F: fun_complex_rat,N: rat,K: nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A2)),K))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),aa_nat_rat(power_power_rat(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8303_prod__le__power,axiom,
    ! [A2: set_nat,F: fun_nat_rat,N: rat,K: nat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),K))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),aa_nat_rat(power_power_rat(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8304_prod__le__power,axiom,
    ! [A2: set_int,F: fun_int_rat,N: rat,K: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A2)),K))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),aa_nat_rat(power_power_rat(N),K))) ) ) ) ).

% prod_le_power
tff(fact_8305_prod__le__power,axiom,
    ! [A2: set_real,F: fun_real_nat,N: nat,K: nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A2)),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ) ) ).

% prod_le_power
tff(fact_8306_prod__le__power,axiom,
    ! [A2: set_complex,F: fun_complex_nat,N: nat,K: nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A2)),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ) ) ).

% prod_le_power
tff(fact_8307_suminf__zero,axiom,
    suminf_real(aTP_Lamm_dz) = zero_zero_real ).

% suminf_zero
tff(fact_8308_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_aj) = zero_zero_nat ).

% suminf_zero
tff(fact_8309_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ea) = zero_zero_int ).

% suminf_zero
tff(fact_8310_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_eb),A2) = one_one_nat ).

% prod.neutral_const
tff(fact_8311_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_ec,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8312_prod_Oneutral__const,axiom,
    ! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ed,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8313_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_bz(F),A2) ).

% int_prod
tff(fact_8314_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_ce(F),A2) ).

% int_prod
tff(fact_8315_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_bd,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8316_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_bd,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_8317_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_complex,A2: set_complex] :
      ( ( groups3708469109370488835omplex(G,A2) != one_one_complex )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A2)
           => ( aa_complex_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8318_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups713298508707869441omplex(G,A2) != one_one_complex )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8319_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups6464643781859351333omplex(G,A2) != one_one_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8320_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups7440179247065528705omplex(G,A2) != one_one_complex )
     => ~ ! [A3: int] :
            ( member_int(A3,A2)
           => ( aa_int_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8321_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups766887009212190081x_real(G,A2) != one_one_real )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A2)
           => ( aa_complex_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8322_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups1681761925125756287l_real(G,A2) != one_one_real )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8323_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A2: set_nat] :
      ( ( groups129246275422532515t_real(G,A2) != one_one_real )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8324_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups2316167850115554303t_real(G,A2) != one_one_real )
     => ~ ! [A3: int] :
            ( member_int(A3,A2)
           => ( aa_int_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8325_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups225925009352817453ex_rat(G,A2) != one_one_rat )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A2)
           => ( aa_complex_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8326_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups4061424788464935467al_rat(G,A2) != one_one_rat )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8327_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( aa_nat_nat(G,X3) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8328_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( aa_nat_int(G,X3) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8329_prod_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => ( aa_int_int(G,X3) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8330_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),aa_complex_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2))) ) ).

% prod_mono
tff(fact_8331_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ).

% prod_mono
tff(fact_8332_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),aa_nat_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A2)),groups129246275422532515t_real(G,A2))) ) ).

% prod_mono
tff(fact_8333_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ).

% prod_mono
tff(fact_8334_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2))) ) ).

% prod_mono
tff(fact_8335_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2))) ) ).

% prod_mono
tff(fact_8336_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),groups73079841787564623at_rat(G,A2))) ) ).

% prod_mono
tff(fact_8337_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(G,A2))) ) ).

% prod_mono
tff(fact_8338_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,A2))) ) ).

% prod_mono
tff(fact_8339_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2))) ) ).

% prod_mono
tff(fact_8340_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_nonneg
tff(fact_8341_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_nonneg
tff(fact_8342_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_nonneg
tff(fact_8343_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_pos
tff(fact_8344_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_pos
tff(fact_8345_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_pos
tff(fact_8346_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8347_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8348_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8349_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_8350_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_8351_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_8352_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_8353_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_8354_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_complex_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups861055069439313189ex_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_8355_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_8356_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8357_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8358_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups129246275422532515t_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8359_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_8360_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X3)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8361_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X3)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8362_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8363_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_8364_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X3)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_8365_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X3)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_8366_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J2))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_8367_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aTP_Lamm_bh(N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bg(C),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8368_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aTP_Lamm_by(X)) ) ).

% summable_arctan_series
tff(fact_8369_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = 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(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_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(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_8370_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ee(I),F)) ).

% summable_single
tff(fact_8371_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ef(I),F)) ).

% summable_single
tff(fact_8372_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_eg(I),F)) ).

% summable_single
tff(fact_8373_summable__zero,axiom,
    summable_real(aTP_Lamm_dz) ).

% summable_zero
tff(fact_8374_summable__zero,axiom,
    summable_nat(aTP_Lamm_aj) ).

% summable_zero
tff(fact_8375_summable__zero,axiom,
    summable_int(aTP_Lamm_ea) ).

% summable_zero
tff(fact_8376_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_eh(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_8377_bit_Oconj__cancel__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = zero_zero_int ).

% bit.conj_cancel_right
tff(fact_8378_bit_Oconj__cancel__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = zero_zero_int ).

% bit.conj_cancel_left
tff(fact_8379_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ei(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8380_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ej(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_8381_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_8382_bit_Ocompl__one,axiom,
    bit_ri7632146776885996613nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_8383_bit_Ocompl__one,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% bit.compl_one
tff(fact_8384_bit_Ocompl__zero,axiom,
    bit_ri7632146776885996613nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_8385_bit_Ocompl__zero,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.compl_zero
tff(fact_8386_bit_Oxor__one__left,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),X) = bit_ri7632146776885996613nteger(X) ).

% bit.xor_one_left
tff(fact_8387_bit_Oxor__one__left,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_left
tff(fact_8388_bit_Oxor__one__right,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(X) ).

% bit.xor_one_right
tff(fact_8389_bit_Oxor__one__right,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_right
tff(fact_8390_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(bit_ri7632146776885996613nteger(X),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_left
tff(fact_8391_bit_Oxor__cancel__left,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_left
tff(fact_8392_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(X,bit_ri7632146776885996613nteger(X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_right
tff(fact_8393_bit_Oxor__cancel__right,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_right
tff(fact_8394_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_8395_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_8396_minus__not__numeral__eq,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_ri7632146776885996613nteger(aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N)) ).

% minus_not_numeral_eq
tff(fact_8397_minus__not__numeral__eq,axiom,
    ! [N: num] : aa_int_int(uminus_uminus_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(inc,N)) ).

% minus_not_numeral_eq
tff(fact_8398_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] : bit_se7788150548672797655nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(aa_nat_Code_integer(bit_se2119862282449309892nteger,N)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_8399_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_8400_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(power_power_real(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_8401_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_8402_not__one__eq,axiom,
    bit_ri7632146776885996613nteger(one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_8403_not__one__eq,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_8404_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(F),G)) ) ) ).

% summable_add
tff(fact_8405_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(F),G)) ) ) ).

% summable_add
tff(fact_8406_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_en(F),G)) ) ) ).

% summable_add
tff(fact_8407_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(F),G)) ) ) ).

% summable_diff
tff(fact_8408_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aTP_Lamm_ep(C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8409_of__int__not__eq,axiom,
    ! [K: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_ri7919022796975470100ot_int,K)) = bit_ri7632146776885996613nteger(aa_int_Code_integer(ring_18347121197199848620nteger,K)) ).

% of_int_not_eq
tff(fact_8410_of__int__not__eq,axiom,
    ! [K: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,K)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(ring_1_of_int_int,K)) ).

% of_int_not_eq
tff(fact_8411_summable__minus__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_eq(F))
    <=> summable_real(F) ) ).

% summable_minus_iff
tff(fact_8412_summable__minus__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_er(F))
    <=> summable_complex(F) ) ).

% summable_minus_iff
tff(fact_8413_summable__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => summable_real(aTP_Lamm_eq(F)) ) ).

% summable_minus
tff(fact_8414_summable__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_complex(aTP_Lamm_er(F)) ) ).

% summable_minus
tff(fact_8415_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_cx(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_8416_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_eh(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_8417_of__int__not__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K))) = bit_ri7632146776885996613nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)) ).

% of_int_not_numeral
tff(fact_8418_of__int__not__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_not_numeral
tff(fact_8419_not__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B) ).

% not_add_distrib
tff(fact_8420_not__diff__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_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(bit_ri7919022796975470100ot_int,A)),B) ).

% not_diff_distrib
tff(fact_8421_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ei(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8422_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_8423_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_8424_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_8425_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_el(F),G)) ) ) ) ).

% suminf_add
tff(fact_8426_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_em(F),G)) ) ) ) ).

% suminf_add
tff(fact_8427_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_en(F),G)) ) ) ) ).

% suminf_add
tff(fact_8428_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_eo(F),G)) ) ) ) ).

% suminf_diff
tff(fact_8429_suminf__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_eq(F)) = aa_real_real(uminus_uminus_real,suminf_real(F)) ) ) ).

% suminf_minus
tff(fact_8430_suminf__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_er(F)) = aa_complex_complex(uminus1482373934393186551omplex,suminf_complex(F)) ) ) ).

% suminf_minus
tff(fact_8431_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N4: nat] : aa_nat_real(F,N4) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8432_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N4: nat] : aa_nat_nat(F,N4) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8433_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N4: nat] : aa_nat_int(F,N4) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8434_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_8435_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_8436_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_8437_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_8438_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_8439_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_8440_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,bit_ri7632146776885996613nteger(A)),one_one_Code_integer) ).

% minus_eq_not_plus_1
tff(fact_8441_minus__eq__not__plus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).

% minus_eq_not_plus_1
tff(fact_8442_minus__eq__not__minus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = bit_ri7632146776885996613nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),one_one_Code_integer)) ).

% minus_eq_not_minus_1
tff(fact_8443_minus__eq__not__minus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ).

% minus_eq_not_minus_1
tff(fact_8444_not__eq__complement,axiom,
    ! [A: code_integer] : bit_ri7632146776885996613nteger(A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),one_one_Code_integer) ).

% not_eq_complement
tff(fact_8445_not__eq__complement,axiom,
    ! [A: int] : aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ).

% not_eq_complement
tff(fact_8446_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_cp(F)) ).

% summable_0_powser
tff(fact_8447_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_cq(F)) ).

% summable_0_powser
tff(fact_8448_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_cp(F)) ).

% summable_zero_power'
tff(fact_8449_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_cq(F)) ).

% summable_zero_power'
tff(fact_8450_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_es(F)) ).

% summable_zero_power'
tff(fact_8451_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_et(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8452_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ev(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8453_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_et(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8454_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ev(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8455_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_8456_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ex(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8457_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_ey(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8458_and__not__numerals_I1_J,axiom,
    aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_8459_disjunctive__diff,axiom,
    ! [B: int,A: int] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(bit_se1146084159140164899it_int(B),N2))
         => pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N2)) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_ri7919022796975470100ot_int,B)) ) ) ).

% disjunctive_diff
tff(fact_8460_take__bit__not__eq__mask__diff,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_int_int(bit_se2923211474154528505it_int(N),A)) ).

% take_bit_not_eq_mask_diff
tff(fact_8461_minus__numeral__inc__eq,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) = bit_ri7632146776885996613nteger(aa_num_Code_integer(numera6620942414471956472nteger,N)) ).

% minus_numeral_inc_eq
tff(fact_8462_minus__numeral__inc__eq,axiom,
    ! [N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)) ).

% minus_numeral_inc_eq
tff(fact_8463_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_8464_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_8465_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_8466_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_8467_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_8468_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_8469_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_8470_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_8471_not__numeral__Bit0__eq,axiom,
    ! [N: num] : bit_ri7632146776885996613nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N))) ).

% not_numeral_Bit0_eq
tff(fact_8472_not__numeral__Bit0__eq,axiom,
    ! [N: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% not_numeral_Bit0_eq
tff(fact_8473_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real))
     => summable_real(power_power_real(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8474_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real))
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8475_summable__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => summable_real(power_power_real(C)) ) ).

% summable_geometric
tff(fact_8476_summable__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_8477_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_cx(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8478_summable__exp,axiom,
    ! [X: complex] : summable_complex(aTP_Lamm_ez(X)) ).

% summable_exp
tff(fact_8479_summable__exp,axiom,
    ! [X: real] : summable_real(aTP_Lamm_fa(X)) ).

% summable_exp
tff(fact_8480_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),N))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N)) ) ).

% bit_minus_int_iff
tff(fact_8481_not__numeral__BitM__eq,axiom,
    ! [N: num] : bit_ri7632146776885996613nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bitM,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N))) ).

% not_numeral_BitM_eq
tff(fact_8482_not__numeral__BitM__eq,axiom,
    ! [N: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bitM,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% not_numeral_BitM_eq
tff(fact_8483_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N))) = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
tff(fact_8484_push__bit__mask__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(bit_se545348938243370406it_int(M),aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,M))) ).

% push_bit_mask_eq
tff(fact_8485_unset__bit__eq__and__not,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(N),A) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).

% unset_bit_eq_and_not
tff(fact_8486_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).

% unset_bit_int_def
tff(fact_8487_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_8488_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_8489_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_et(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8490_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ev(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8491_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_et(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8492_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ev(F),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8493_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N7: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_eh(F),N6)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_8494_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N7: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_fb(F),N6)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_8495_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real))
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ew(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8496_bit__not__iff__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,A)),N))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
        & ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(A),N)) ) ) ).

% bit_not_iff_eq
tff(fact_8497_minus__exp__eq__not__mask,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N)) = bit_ri7632146776885996613nteger(aa_nat_Code_integer(bit_se2119862282449309892nteger,N)) ).

% minus_exp_eq_not_mask
tff(fact_8498_minus__exp__eq__not__mask,axiom,
    ! [N: nat] : aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) ).

% minus_exp_eq_not_mask
tff(fact_8499_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8500_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8501_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% and_not_numerals(8)
tff(fact_8502_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_8503_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = 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(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_vebt_buildup(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_vebt_buildup(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))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = 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,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_vebt_buildup(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_vebt_buildup(aa_nat_nat(suc,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_buildup.elims
tff(fact_8504_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aTP_Lamm_fc(X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_8505_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_8506_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_8507_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(power_power_real(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_real )
           => ( groups6591440286371151544t_real(power_power_real(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_real )
           => ( groups6591440286371151544t_real(power_power_real(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N)))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_8508_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa3))
               => ( ( ( Xa3 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => pp(B3) )
                      & ( Xa3 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = 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),Xa3)) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = 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),Xa3)) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = 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),Xa3)) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa3))
                       => ( ( Xa3 != Mi )
                         => ( ( Xa3 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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(vEBT_VEBT_low,Xa3),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)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_8509_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X)),I))) ) ) ).

% intind
tff(fact_8510_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_bool,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_int_bool(P,X))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(replicate_int(N,X)),I))) ) ) ).

% intind
tff(fact_8511_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I))) ) ) ).

% intind
tff(fact_8512_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_8513_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_int(aa_complex_int(F,X3),ring_1_Ints_int) )
     => member_int(groups5690904116761175830ex_int(F,A2),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_8514_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => member_int(aa_real_int(F,X3),ring_1_Ints_int) )
     => member_int(groups1932886352136224148al_int(F,A2),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_8515_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => member_int(aa_nat_int(F,X3),ring_1_Ints_int) )
     => member_int(groups3539618377306564664at_int(F,A2),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_8516_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_real(aa_complex_real(F,X3),ring_1_Ints_real) )
     => member_real(groups5808333547571424918x_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_8517_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => member_real(aa_real_real(F,X3),ring_1_Ints_real) )
     => member_real(groups8097168146408367636l_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_8518_Ints__sum,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => member_real(aa_int_real(F,X3),ring_1_Ints_real) )
     => member_real(groups8778361861064173332t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_8519_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => member_complex(aa_real_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups5754745047067104278omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8520_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => member_complex(aa_nat_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups2073611262835488442omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8521_Ints__sum,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => member_complex(aa_int_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups3049146728041665814omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8522_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_complex(aa_complex_complex(F,X3),ring_1_Ints_complex) )
     => member_complex(groups7754918857620584856omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8523_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_8524_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_fd,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_8525_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_ak,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_8526_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_dz,A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_8527_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_fe(F),A2) ).

% of_nat_sum
tff(fact_8528_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_bz(F),A2) ).

% of_nat_sum
tff(fact_8529_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_ce(F),A2) ).

% of_nat_sum
tff(fact_8530_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups7501900531339628137nteger(aTP_Lamm_cb(F),A2) ).

% of_nat_sum
tff(fact_8531_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups2906978787729119204at_rat(aTP_Lamm_cc(F),A2) ).

% of_nat_sum
tff(fact_8532_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd(F)),A2) ).

% of_nat_sum
tff(fact_8533_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups6591440286371151544t_real(aTP_Lamm_ca(F),A2) ).

% of_nat_sum
tff(fact_8534_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_ff(F),A2) ).

% of_int_sum
tff(fact_8535_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = groups6591440286371151544t_real(aTP_Lamm_cf(F),A2) ).

% of_int_sum
tff(fact_8536_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_ck(F),A2) ).

% of_int_sum
tff(fact_8537_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_Code_integer(ring_18347121197199848620nteger,groups4538972089207619220nt_int(F,A2)) = groups7873554091576472773nteger(aTP_Lamm_cl(F),A2) ).

% of_int_sum
tff(fact_8538_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_cm(F),A2) ).

% of_int_sum
tff(fact_8539_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups4538972089207619220nt_int(F,A2)) = groups3049146728041665814omplex(aTP_Lamm_cn(F),A2) ).

% of_int_sum
tff(fact_8540_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_co(F),A2) ).

% of_int_sum
tff(fact_8541_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_dz),zero_zero_real)) ).

% sums_zero
tff(fact_8542_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_aj),zero_zero_nat)) ).

% sums_zero
tff(fact_8543_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_ea),zero_zero_int)) ).

% sums_zero
tff(fact_8544_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_8545_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_8546_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_8547_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_8548_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_8549_sum_Oempty,axiom,
    ! [G: fun_nat_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),bot_bot_set_nat) = zero_zero_nat ).

% sum.empty
tff(fact_8550_sum_Oempty,axiom,
    ! [G: fun_complex_complex] : groups7754918857620584856omplex(G,bot_bot_set_complex) = zero_zero_complex ).

% sum.empty
tff(fact_8551_sum_Oempty,axiom,
    ! [G: fun_int_int] : groups4538972089207619220nt_int(G,bot_bot_set_int) = zero_zero_int ).

% sum.empty
tff(fact_8552_sum_Oempty,axiom,
    ! [G: fun_nat_real] : groups6591440286371151544t_real(G,bot_bot_set_nat) = zero_zero_real ).

% sum.empty
tff(fact_8553_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X5)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8554_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( member_int(X5,set_int2(replicate_int(N,A)))
         => pp(aa_int_bool(P,X5)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8555_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8556_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X5)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8557_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ? [X5: int] :
          ( member_int(X5,set_int2(replicate_int(N,A)))
          & pp(aa_int_bool(P,X5)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8558_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8559_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( member_complex(X,set_complex2(replicate_complex(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8560_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( member_real(X,set_real2(replicate_real(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8561_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( member_set_nat(X,set_set_nat2(replicate_set_nat(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8562_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8563_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( member_int(X,set_int2(replicate_int(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8564_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8565_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_8566_nth__replicate,axiom,
    ! [I: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_8567_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I) = X ) ) ).

% nth_replicate
tff(fact_8568_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_fg(F),A2))) ).

% sum_abs_ge_zero
tff(fact_8569_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_fh(F),A2))) ).

% sum_abs_ge_zero
tff(fact_8570_sum__constant,axiom,
    ! [Y: int,A2: set_complex] : groups5690904116761175830ex_int(aTP_Lamm_fi(Y),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_complex(A2))),Y) ).

% sum_constant
tff(fact_8571_sum__constant,axiom,
    ! [Y: int,A2: set_nat] : groups3539618377306564664at_int(aTP_Lamm_fj(Y),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A2))),Y) ).

% sum_constant
tff(fact_8572_sum__constant,axiom,
    ! [Y: real,A2: set_complex] : groups5808333547571424918x_real(aTP_Lamm_fk(Y),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A2))),Y) ).

% sum_constant
tff(fact_8573_sum__constant,axiom,
    ! [Y: real,A2: set_int] : groups8778361861064173332t_real(aTP_Lamm_fl(Y),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A2))),Y) ).

% sum_constant
tff(fact_8574_sum__constant,axiom,
    ! [Y: nat,A2: set_complex] : groups5693394587270226106ex_nat(aTP_Lamm_fm(Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_complex(A2))),Y) ).

% sum_constant
tff(fact_8575_sum__constant,axiom,
    ! [Y: nat,A2: set_int] : groups4541462559716669496nt_nat(aTP_Lamm_fn(Y),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A2))),Y) ).

% sum_constant
tff(fact_8576_sum__constant,axiom,
    ! [Y: code_integer,A2: set_complex] : groups6621422865394947399nteger(aTP_Lamm_fo(Y),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_complex(A2))),Y) ).

% sum_constant
tff(fact_8577_sum__constant,axiom,
    ! [Y: code_integer,A2: set_nat] : groups7501900531339628137nteger(aTP_Lamm_fp(Y),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A2))),Y) ).

% sum_constant
tff(fact_8578_sum__constant,axiom,
    ! [Y: code_integer,A2: set_int] : groups7873554091576472773nteger(aTP_Lamm_fq(Y),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A2))),Y) ).

% sum_constant
tff(fact_8579_sum__constant,axiom,
    ! [Y: rat,A2: set_complex] : groups5058264527183730370ex_rat(aTP_Lamm_fr(Y),A2) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A2))),Y) ).

% sum_constant
tff(fact_8580_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_8581_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_8582_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_8583_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_cp(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8584_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_cq(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8585_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8586_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8587_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8588_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8589_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ee(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_8590_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ef(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_8591_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_eg(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_8592_sums__diff,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(F),G)),aa_real_real(minus_minus_real(A),B))) ) ) ).

% sums_diff
tff(fact_8593_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% sums_add
tff(fact_8594_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( pp(aa_nat_bool(sums_nat(F),A))
     => ( pp(aa_nat_bool(sums_nat(G),B))
       => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% sums_add
tff(fact_8595_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( pp(aa_int_bool(sums_int(F),A))
     => ( pp(aa_int_bool(sums_int(G),B))
       => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_en(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% sums_add
tff(fact_8596_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N2: nat] : aa_nat_real(F,N2) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_8597_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_nat(F,N2) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_8598_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N2: nat] : aa_nat_int(F,N2) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_8599_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups5808333547571424918x_real(G,A2) != zero_zero_real )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A2)
           => ( aa_complex_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8600_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8601_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( member_int(A3,A2)
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8602_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups5058264527183730370ex_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A2)
           => ( aa_complex_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8603_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8604_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8605_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( member_int(A3,A2)
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8606_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_nat,A2: set_complex] :
      ( ( groups5693394587270226106ex_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A2)
           => ( aa_complex_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8607_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A2: set_real] :
      ( ( groups1935376822645274424al_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8608_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A2: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: int] :
            ( member_int(A3,A2)
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8609_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( aa_nat_nat(G,X3) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_8610_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => ( aa_complex_complex(G,X3) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_8611_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => ( aa_int_int(G,X3) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_8612_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( aa_nat_real(G,X3) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_8613_sums__minus,axiom,
    ! [F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => pp(aa_real_bool(sums_real(aTP_Lamm_eq(F)),aa_real_real(uminus_uminus_real,A))) ) ).

% sums_minus
tff(fact_8614_sums__minus,axiom,
    ! [F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(F),A))
     => pp(aa_complex_bool(sums_complex(aTP_Lamm_er(F)),aa_complex_complex(uminus1482373934393186551omplex,A))) ) ).

% sums_minus
tff(fact_8615_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_8616_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fs(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_8617_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ft(G),H),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_8618_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).

% sum.distrib
tff(fact_8619_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fu(F),G),A2) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,A2)) ).

% sum_subtractf
tff(fact_8620_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fv(F),G),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ).

% sum_subtractf
tff(fact_8621_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(F),G),A2) = aa_real_real(minus_minus_real(groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,A2)) ).

% sum_subtractf
tff(fact_8622_sum__negf,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_fw(F),A2) = aa_complex_complex(uminus1482373934393186551omplex,groups7754918857620584856omplex(F,A2)) ).

% sum_negf
tff(fact_8623_sum__negf,axiom,
    ! [F: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_fx(F),A2) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(F,A2)) ).

% sum_negf
tff(fact_8624_sum__negf,axiom,
    ! [F: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_eq(F),A2) = aa_real_real(uminus_uminus_real,groups6591440286371151544t_real(F,A2)) ).

% sum_negf
tff(fact_8625_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A2))) ) ).

% sum_nonneg
tff(fact_8626_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2))) ) ).

% sum_nonneg
tff(fact_8627_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2))) ) ).

% sum_nonneg
tff(fact_8628_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8629_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8630_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8631_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8632_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_8633_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_8634_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_8635_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X3)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8636_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X3)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8637_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X3)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8638_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8639_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8640_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8641_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8642_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X3)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8643_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X3)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8644_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8645_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X3)),A2))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X3)) = aa_nat_nat(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8646_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X3)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X3)) = aa_nat_real(G,aa_nat_nat(suc,X3)) ) )
       => ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8647_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ei(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_8648_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fy(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_8649_replicate__eqI,axiom,
    ! [Xs: list_complex,N: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = N )
     => ( ! [Y3: complex] :
            ( member_complex(Y3,set_complex2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_complex(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8650_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y3: real] :
            ( member_real(Y3,set_real2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8651_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y3: set_nat] :
            ( member_set_nat(Y3,set_set_nat2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_set_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8652_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),set_VEBT_VEBT2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8653_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y3: bool] :
            ( member_bool(Y3,set_bool2(Xs))
           => ( pp(Y3)
            <=> pp(X) ) )
       => ( Xs = replicate_bool(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8654_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = N )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8655_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y3: int] :
            ( member_int(Y3,set_int2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(N,X) ) ) ) ).

% replicate_eqI
tff(fact_8656_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => ( X3 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8657_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => ( pp(X3)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8658_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => ( X3 = X ) )
     => ( replicate_nat(aa_list_nat_nat(size_size_list_nat,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8659_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => ( X3 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8660_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cr(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8661_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_cx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8662_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ct(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8663_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_eh(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8664_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fz(C),F)),A))
     => ( ( C != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ) ).

% sums_mult_D
tff(fact_8665_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ei(C),F)),A))
     => ( ( C != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(divide_divide_real(A),C))) ) ) ).

% sums_mult_D
tff(fact_8666_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_cx(F)),S))
       => pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_8667_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_cx(F)),L))
     => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_8668_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_cr(F)),L))
     => pp(aa_nat_bool(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_8669_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_cs(F)),L))
     => pp(aa_int_bool(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_8670_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_cx(F)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_8671_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(N),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_8672_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_gb(X),M),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),I5)) ).

% sum_power_add
tff(fact_8673_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_gc(X),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_8674_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_gd(X),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_8675_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ge(X),M),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),groups6591440286371151544t_real(power_power_real(X),I5)) ).

% sum_power_add
tff(fact_8676_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_cv(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8677_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gf(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8678_sum__bounded__above,axiom,
    ! [A2: set_real,F: fun_real_real,K5: real] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8679_sum__bounded__above,axiom,
    ! [A2: set_complex,F: fun_complex_real,K5: real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8680_sum__bounded__above,axiom,
    ! [A2: set_int,F: fun_int_real,K5: real] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8681_sum__bounded__above,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger,K5: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_real_Code_integer(F,I2)),K5)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7713935264441627589nteger(F,A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_real(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8682_sum__bounded__above,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger,K5: code_integer] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F,I2)),K5)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups6621422865394947399nteger(F,A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_complex(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8683_sum__bounded__above,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,K5: code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I2)),K5)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8684_sum__bounded__above,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,K5: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I2)),K5)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8685_sum__bounded__above,axiom,
    ! [A2: set_real,F: fun_real_rat,K5: rat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8686_sum__bounded__above,axiom,
    ! [A2: set_complex,F: fun_complex_rat,K5: rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8687_sum__bounded__above,axiom,
    ! [A2: set_nat,F: fun_nat_rat,K5: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A2))),K5))) ) ).

% sum_bounded_above
tff(fact_8688_sum__bounded__below,axiom,
    ! [A2: set_real,K5: real,F: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K5),aa_real_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A2))),K5)),groups8097168146408367636l_real(F,A2))) ) ).

% sum_bounded_below
tff(fact_8689_sum__bounded__below,axiom,
    ! [A2: set_complex,K5: real,F: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K5),aa_complex_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A2))),K5)),groups5808333547571424918x_real(F,A2))) ) ).

% sum_bounded_below
tff(fact_8690_sum__bounded__below,axiom,
    ! [A2: set_int,K5: real,F: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K5),aa_int_real(F,I2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A2))),K5)),groups8778361861064173332t_real(F,A2))) ) ).

% sum_bounded_below
tff(fact_8691_sum__bounded__below,axiom,
    ! [A2: set_real,K5: code_integer,F: fun_re6978948838062957830nteger] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K5),aa_real_Code_integer(F,I2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_real(A2))),K5)),groups7713935264441627589nteger(F,A2))) ) ).

% sum_bounded_below
tff(fact_8692_sum__bounded__below,axiom,
    ! [A2: set_complex,K5: code_integer,F: fun_co8153928105989009288nteger] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K5),aa_com4217951221296830281nteger(F,I2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_complex(A2))),K5)),groups6621422865394947399nteger(F,A2))) ) ).

% sum_bounded_below
tff(fact_8693_sum__bounded__below,axiom,
    ! [A2: set_nat,K5: code_integer,F: fun_nat_Code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K5),aa_nat_Code_integer(F,I2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A2))),K5)),groups7501900531339628137nteger(F,A2))) ) ).

% sum_bounded_below
tff(fact_8694_sum__bounded__below,axiom,
    ! [A2: set_int,K5: code_integer,F: fun_int_Code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K5),aa_int_Code_integer(F,I2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A2))),K5)),groups7873554091576472773nteger(F,A2))) ) ).

% sum_bounded_below
tff(fact_8695_sum__bounded__below,axiom,
    ! [A2: set_real,K5: rat,F: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_real_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A2))),K5)),groups1300246762558778688al_rat(F,A2))) ) ).

% sum_bounded_below
tff(fact_8696_sum__bounded__below,axiom,
    ! [A2: set_complex,K5: rat,F: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_complex_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A2))),K5)),groups5058264527183730370ex_rat(F,A2))) ) ).

% sum_bounded_below
tff(fact_8697_sum__bounded__below,axiom,
    ! [A2: set_nat,K5: rat,F: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_nat_rat(F,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A2))),K5)),groups2906978787729119204at_rat(F,A2))) ) ).

% sum_bounded_below
tff(fact_8698_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_gg(M),Z)),aa_nat_complex(power_power_complex(Z),M))) ).

% powser_sums_if
tff(fact_8699_powser__sums__if,axiom,
    ! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_gh(M),Z)),aa_nat_real(power_power_real(Z),M))) ).

% powser_sums_if
tff(fact_8700_powser__sums__if,axiom,
    ! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_gi(M),Z)),aa_nat_int(power_power_int(Z),M))) ).

% powser_sums_if
tff(fact_8701_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_cp(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_8702_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_cq(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_8703_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8704_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8705_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8706_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8707_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8708_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8709_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8710_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8711_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8712_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8713_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8714_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8715_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,aa_nat_nat(suc,N))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8716_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8717_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8718_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8719_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_8720_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_8721_set__replicate__Suc,axiom,
    ! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_8722_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_8723_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_8724_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_8725_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_cy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8726_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_cs(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8727_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cr(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8728_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(aTP_Lamm_cx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8729_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_gj(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8730_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(aTP_Lamm_gk(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8731_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6591440286371151544t_real(aTP_Lamm_gl(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(minus_minus_real(aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8732_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_gm(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8733_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_gn(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8734_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_go(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8735_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_gp(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8736_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8737_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8738_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8739_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8740_sum__bounded__above__strict,axiom,
    ! [A2: set_real,F: fun_real_int,K5: int] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F,A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_real(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8741_sum__bounded__above__strict,axiom,
    ! [A2: set_complex,F: fun_complex_int,K5: int] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_complex(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8742_sum__bounded__above__strict,axiom,
    ! [A2: set_nat,F: fun_nat_int,K5: int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8743_sum__bounded__above__strict,axiom,
    ! [A2: set_real,F: fun_real_real,K5: real] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8744_sum__bounded__above__strict,axiom,
    ! [A2: set_complex,F: fun_complex_real,K5: real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8745_sum__bounded__above__strict,axiom,
    ! [A2: set_int,F: fun_int_real,K5: real] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8746_sum__bounded__above__strict,axiom,
    ! [A2: set_real,F: fun_real_nat,K5: nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_real(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8747_sum__bounded__above__strict,axiom,
    ! [A2: set_complex,F: fun_complex_nat,K5: nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_complex(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8748_sum__bounded__above__strict,axiom,
    ! [A2: set_int,F: fun_int_nat,K5: nat] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8749_sum__bounded__above__strict,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger,K5: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_real_Code_integer(F,I2)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A2)))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups7713935264441627589nteger(F,A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_real(A2))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_8750_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I2))) )
     => ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( member_complex(I2,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_com4217951221296830281nteger(A,I2)),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_gq(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8751_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I2))) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_real_Code_integer(A,I2)),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_gr(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8752_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2))) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(A,I2)),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_gs(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8753_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2))) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(A,I2)),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_gt(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8754_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(X,I2))) )
     => ( ( groups5808333547571424918x_real(X,I5) = one_one_real )
       => ( ! [I2: complex] :
              ( member_complex(I2,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(A,I2)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gu(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8755_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I2))) )
     => ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(A,I2)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8756_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I2))) )
     => ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(A,I2)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gw(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8757_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X,I2))) )
     => ( ( groups5058264527183730370ex_rat(X,I5) = one_one_rat )
       => ( ! [I2: complex] :
              ( member_complex(I2,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gx(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8758_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I2))) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gy(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8759_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I2))) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gz(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8760_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_ha(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_ha(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_8761_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_hb(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_hb(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_8762_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups6591440286371151544t_real(aTP_Lamm_hc(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(minus_minus_real(aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups6591440286371151544t_real(aTP_Lamm_hc(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_8763_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_hd(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_8764_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_he(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_8765_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(aTP_Lamm_hf(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_real_real(minus_minus_real(aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_8766_geometric__sums,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => pp(aa_real_bool(sums_real(power_power_real(C)),aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C)))) ) ).

% geometric_sums
tff(fact_8767_geometric__sums,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => pp(aa_complex_bool(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)))) ) ).

% geometric_sums
tff(fact_8768_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_hg),one_one_real)) ).

% power_half_series
tff(fact_8769_mask__eq__sum__exp,axiom,
    ! [N: nat] : 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))),N)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N))) ).

% mask_eq_sum_exp
tff(fact_8770_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N))) ).

% mask_eq_sum_exp
tff(fact_8771_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8772_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8773_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8774_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(one_one_real),X)),groups6591440286371151544t_real(power_power_real(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8775_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_hh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8776_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_hi(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8777_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hj(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8778_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups6591440286371151544t_real(aTP_Lamm_hk(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8779_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => pp(aa_real_bool(sums_real(aTP_Lamm_hl(G)),X)) ) ).

% sums_if'
tff(fact_8780_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => ( pp(aa_real_bool(sums_real(F),Y))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hm(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sums_if
tff(fact_8781_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N))) ).

% mask_eq_sum_exp_nat
tff(fact_8782_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8783_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2073611262835488442omplex(aTP_Lamm_hn(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8784_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_ho(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8785_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups6591440286371151544t_real(aTP_Lamm_hp(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8786_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_hq(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).

% double_arith_series
tff(fact_8787_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_8788_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_hs(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D))) ).

% double_arith_series
tff(fact_8789_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ht(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_8790_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_8791_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hv(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).

% double_arith_series
tff(fact_8792_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_8793_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_8794_double__gauss__sum,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_8795_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8796_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_8797_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_8798_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hw(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8799_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8800_arith__series,axiom,
    ! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8801_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_hs(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8802_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8803_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8804_gauss__sum,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8805_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8806_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8807_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8808_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8809_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8810_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8811_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8812_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8813_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8814_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( groups6591440286371151544t_real(power_power_real(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( groups6591440286371151544t_real(power_power_real(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N))))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8815_cos__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aTP_Lamm_hx(X)),aa_real_real(cos_real,X))) ).

% cos_paired
tff(fact_8816_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = 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(aa_nat_fun_nat_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_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_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_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = 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,aa_nat_nat(aa_nat_fun_nat_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_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_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_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_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_buildup.simps(3)
tff(fact_8817_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8818_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8819_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8820_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_hy(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8821_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_hz(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8822_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_ia(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8823_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa3))
               => ~ ( ( ( Xa3 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => pp(B3) )
                        & ( Xa3 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa3))
                 => ~ ( ( Xa3 != Mi )
                     => ( ( Xa3 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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(vEBT_VEBT_low,Xa3),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)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_8824_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => pp(aa_real_bool(sums_real(aTP_Lamm_ib(Z)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_8825_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aTP_Lamm_ic(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_8826_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa3))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa3 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => pp(B3) )
                        & ( Xa3 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa3)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa3)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xa3)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(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),Xa3)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa3 != Mi )
                           => ( ( Xa3 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),Mi))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa3))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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(vEBT_VEBT_low,Xa3),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)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa3)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_8827_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa3))
               => ( ( ( Xa3 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => pp(B3) )
                      & ( Xa3 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa3)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2),Xa3))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_8828_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa3))
               => ~ ( ( ( Xa3 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => pp(B3) )
                        & ( Xa3 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2),Xa3))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_8829_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa3))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa3 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => pp(B3) )
                        & ( Xa3 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa3)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa3)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S2),Xa3)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_8830_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa3)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa3)) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),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(Mi),Ma2)),zero_zero_nat,Va3,Vb2),Xa3))
                   => ( ( Xa3 = Mi )
                      | ( Xa3 = Ma2 ) ) ) )
             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2),Xa3))
                     => ( ( Xa3 = Mi )
                        | ( Xa3 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
               => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd),Xa3))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_8831_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_bz(F),A2) ).

% int_sum
tff(fact_8832_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_ce(F),A2) ).

% int_sum
tff(fact_8833_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(G,X3)),aa_complex_nat(F,X3))) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_id(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8834_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X3)),aa_real_nat(F,X3))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ie(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8835_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G,X3)),aa_set_nat_nat(F,X3))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_if(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8836_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X3: int] :
          ( member_int(X3,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ig(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8837_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ih(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8838_card__eq__sum,axiom,
    ! [A2: set_complex] : finite_card_complex(A2) = groups5693394587270226106ex_nat(aTP_Lamm_ii,A2) ).

% card_eq_sum
tff(fact_8839_card__eq__sum,axiom,
    ! [A2: set_list_nat] : finite_card_list_nat(A2) = groups4396056296759096172at_nat(aTP_Lamm_ij,A2) ).

% card_eq_sum
tff(fact_8840_card__eq__sum,axiom,
    ! [A2: set_set_nat] : finite_card_set_nat(A2) = groups8294997508430121362at_nat(aTP_Lamm_ik,A2) ).

% card_eq_sum
tff(fact_8841_card__eq__sum,axiom,
    ! [A2: set_int] : finite_card_int(A2) = groups4541462559716669496nt_nat(aTP_Lamm_il,A2) ).

% card_eq_sum
tff(fact_8842_card__eq__sum,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(finite_card_nat,A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_eb),A2) ).

% card_eq_sum
tff(fact_8843_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N) )
     => ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) ) ) ).

% sum_SucD
tff(fact_8844_sum__Suc,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : groups5693394587270226106ex_nat(aTP_Lamm_im(F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,A2)),finite_card_complex(A2)) ).

% sum_Suc
tff(fact_8845_sum__Suc,axiom,
    ! [F: fun_list_nat_nat,A2: set_list_nat] : groups4396056296759096172at_nat(aTP_Lamm_in(F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(F,A2)),finite_card_list_nat(A2)) ).

% sum_Suc
tff(fact_8846_sum__Suc,axiom,
    ! [F: fun_set_nat_nat,A2: set_set_nat] : groups8294997508430121362at_nat(aTP_Lamm_io(F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(F,A2)),finite_card_set_nat(A2)) ).

% sum_Suc
tff(fact_8847_sum__Suc,axiom,
    ! [F: fun_int_nat,A2: set_int] : groups4541462559716669496nt_nat(aTP_Lamm_ip(F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,A2)),finite_card_int(A2)) ).

% sum_Suc
tff(fact_8848_sum__Suc,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_iq(F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(finite_card_nat,A2)) ).

% sum_Suc
tff(fact_8849_real__of__card,axiom,
    ! [A2: set_complex] : aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A2)) = groups5808333547571424918x_real(aTP_Lamm_ir,A2) ).

% real_of_card
tff(fact_8850_real__of__card,axiom,
    ! [A2: set_list_nat] : aa_nat_real(semiri5074537144036343181t_real,finite_card_list_nat(A2)) = groups8399112307953289288t_real(aTP_Lamm_is,A2) ).

% real_of_card
tff(fact_8851_real__of__card,axiom,
    ! [A2: set_set_nat] : aa_nat_real(semiri5074537144036343181t_real,finite_card_set_nat(A2)) = groups5107569545109728110t_real(aTP_Lamm_it,A2) ).

% real_of_card
tff(fact_8852_real__of__card,axiom,
    ! [A2: set_int] : aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A2)) = groups8778361861064173332t_real(aTP_Lamm_iu,A2) ).

% real_of_card
tff(fact_8853_real__of__card,axiom,
    ! [A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(finite_card_nat,A2)) = groups6591440286371151544t_real(aTP_Lamm_iv,A2) ).

% real_of_card
tff(fact_8854_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: fun_set_nat_nat] :
      ( ( member_set_nat(A,A2)
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)) ) )
      & ( ~ member_set_nat(A,A2)
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = groups8294997508430121362at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_8855_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: fun_int_nat] :
      ( ( member_int(A,A2)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)) ) )
      & ( ~ member_int(A,A2)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_8856_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) ) ) ) ).

% sum_diff1_nat
tff(fact_8857_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_az,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_iw(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8858_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_az,collect_complex(aTP_Lamm_bh(N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8859_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_bd,set_or1266510415728281911st_int(M,N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_8860_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),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(Mi),Ma2)),zero_zero_nat,Va3,Vb2),Xa3))
               => ~ ( ( Xa3 = Mi )
                    | ( Xa3 = Ma2 ) ) ) )
         => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2),Xa3))
                 => ~ ( ( Xa3 = Mi )
                      | ( Xa3 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
           => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd),Xa3))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_8861_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa3))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa3)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa3)) ) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa3 = Mi )
                        | ( Xa3 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2),Xa3)) ) )
             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa3 = Mi )
                          | ( Xa3 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList,Vc2),Xa3)) ) )
               => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa3),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)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa3),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd),Xa3)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_8862_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_ix(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8863_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_ix(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8864_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_iy(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8865_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_iy(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8866_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8867_sumr__cos__zero__one,axiom,
    ! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_iz,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8868_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_8869_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8870_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ja(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))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_jb(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),groups6591440286371151544t_real(aTP_Lamm_jc(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_8871_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,X)),groups6591440286371151544t_real(aTP_Lamm_iy(X),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_8872_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))),D4)))),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))),D4)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_8873_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_jd(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8874_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] : aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_iy(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ).

% Maclaurin_sin_expansion
tff(fact_8875_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_iy(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8876_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_ix(X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8877_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => bij_betw_nat_complex(aTP_Lamm_je(N),aa_nat_set_nat(set_ord_lessThan_nat,N),collect_complex(aTP_Lamm_bh(N))) ) ).

% bij_betw_roots_unity
tff(fact_8878_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_8879_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_8880_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_8881_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_8882_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_8883_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jf,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8884_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jg,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),N))),N) ).

% sum_choose_lower
tff(fact_8885_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8886_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8887_diffs__cos__coeff,axiom,
    ! [X4: nat] : aa_nat_real(diffs_real(cos_coeff),X4) = aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,X4)) ).

% diffs_cos_coeff
tff(fact_8888_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_8889_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8890_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jj(M),N),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R2) ).

% vandermonde
tff(fact_8891_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jk(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_8892_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jl(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jl(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jn(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_8893_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_8894_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jo,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_8895_complex__sgn__def,axiom,
    ! [X: complex] : aa_complex_complex(sgn_sgn_complex,X) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,X))),X) ).

% complex_sgn_def
tff(fact_8896_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_real(arg,Z) = zero_zero_real ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_real(arg,Z) = fChoice_real(aTP_Lamm_jp(Z)) ) ) ) ).

% Arg_def
tff(fact_8897_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_8898_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = 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(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_vebt_buildup(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_vebt_buildup(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))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = 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,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_vebt_buildup(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_vebt_buildup(aa_nat_nat(suc,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)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_8899_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jq(L)),Qr) ).

% divmod_step_nat_def
tff(fact_8900_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_jr(L),Qr) ).

% divmod_step_int_def
tff(fact_8901_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_js),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_8902_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jt,Y)) ).

% arctan_def
tff(fact_8903_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ju,Y)) ).

% arcsin_def
tff(fact_8904_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_8905_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,K)))) ).

% finite_Collect_less_nat
tff(fact_8906_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_jv(A),B))) ).

% finite_interval_int1
tff(fact_8907_finite__interval__int4,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_jw(A),B))) ).

% finite_interval_int4
tff(fact_8908_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_jx(A),B))) ).

% finite_interval_int3
tff(fact_8909_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_jy(A),B))) ).

% finite_interval_int2
tff(fact_8910_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_8911_set__decode__inverse,axiom,
    ! [N: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,N)) = N ).

% set_decode_inverse
tff(fact_8912_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_8913_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => finite3207457112153483333omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_iw(N),C))) ) ).

% finite_nth_roots
tff(fact_8914_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,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))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_8915_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_8916_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_8917_finite__nat__set__iff__bounded,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M4: nat] :
        ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),M4)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_8918_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% bounded_nat_set_is_finite
tff(fact_8919_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(nat_set_decode,N))) ).

% finite_set_decode
tff(fact_8920_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jz(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_8921_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ay,I))) ) ).

% finite_divisors_int
tff(fact_8922_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_8923_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,M)))) ) ).

% finite_divisors_nat
tff(fact_8924_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_8925_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_8926_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_8927_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_8928_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_8929_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_8930_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
         => pp(aa_nat_bool(P,M4)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less_eq
tff(fact_8931_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
          & pp(aa_nat_bool(P,M4)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less_eq
tff(fact_8932_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_8933_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_8934_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_8935_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8936_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_8937_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_8938_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2)))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_8939_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_8940_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_8941_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_8942_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N5)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8943_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),S3))) ).

% card_sum_le_nat_sum
tff(fact_8944_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8945_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_8946_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ka(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),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),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8947_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ! [M4: int] :
        ? [N4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M4),aa_int_int(abs_abs_int,N4)))
          & member_int(N4,S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8948_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_8949_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_finite_int(set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_8950_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_8951_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_8952_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_8953_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M2))
         => ? [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N6))
              & pp(aa_set_nat_bool(member_nat(N6),S3)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S3)) ) ).

% unbounded_k_infinite
tff(fact_8954_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M4: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N4))
          & pp(aa_set_nat_bool(member_nat(N4),S3)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8955_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_kb(D))) ).

% int_ge_less_than2_def
tff(fact_8956_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_kc(D))) ).

% int_ge_less_than_def
tff(fact_8957_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).

% int_of_nat_def
tff(fact_8958_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_kd(L),Qr) ).

% divmod_step_integer_def
tff(fact_8959_or__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(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(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_8960_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_8961_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_8962_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(2)
tff(fact_8963_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(6)
tff(fact_8964_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_8965_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_8966_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_8967_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_8968_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(X),Y))) ) ) ).

% OR_lower
tff(fact_8969_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L))) ) ).

% or_greater_eq
tff(fact_8970_plus__and__or,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),aa_int_int(bit_se1409905431419307370or_int(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) ).

% plus_and_or
tff(fact_8971_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_ke),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_8972_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_kf),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_8973_or__not__numerals_I1_J,axiom,
    aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_8974_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_8975_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_8976_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_8977_set__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(bit_se1409905431419307370or_int(K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% set_bit_int_def
tff(fact_8978_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_8979_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_8980_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_8981_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(2)
tff(fact_8982_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_8983_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(3)
tff(fact_8984_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_8985_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_8986_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_8987_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_8988_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_8989_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_8990_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,N)))) ).

% or_minus_numerals(5)
tff(fact_8991_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,N)))) ).

% or_minus_numerals(1)
tff(fact_8992_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_18347121197199848620nteger ).

% integer_of_int_eq_of_int
tff(fact_8993_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_8994_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_8995_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_8996_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_8997_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_8998_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_8999_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,N)))) ).

% or_minus_numerals(7)
tff(fact_9000_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,N)))) ).

% or_minus_numerals(3)
tff(fact_9001_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = K ) ) ) ).

% abs_integer_code
tff(fact_9002_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_9003_uminus__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,X)) ).

% uminus_integer.abs_eq
tff(fact_9004_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9005_less__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa3)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa3),X)) ) ).

% less_integer.abs_eq
tff(fact_9006_plus__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(code_integer_of_int,Xa3)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X)) ).

% plus_integer.abs_eq
tff(fact_9007_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_9008_minus__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(code_integer_of_int,Xa3)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X)) ).

% minus_integer.abs_eq
tff(fact_9009_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9010_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% or_nat_def
tff(fact_9011_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),N))) ).

% int_numeral_or_not_num_neg
tff(fact_9012_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(N),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9013_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),N)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% numeral_or_not_num_eq
tff(fact_9014_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aTP_Lamm_kg(X)) ).

% floor_real_def
tff(fact_9015_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_9016_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_9017_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9018_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K = zero_zero_int )
           => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = K ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = 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(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_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(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_9019_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_kh(X)) ).

% floor_rat_def
tff(fact_9020_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = 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,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(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_9021_integer__of__num_I3_J,axiom,
    ! [N: num] : code_integer_of_num(aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,code_integer_of_num(N)),code_integer_of_num(N))),one_one_Code_integer) ).

% integer_of_num(3)
tff(fact_9022_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_9023_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_9024_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_9025_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_9026_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_9027_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_9028_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_9029_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_9030_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9031_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_rat_def
tff(fact_9032_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ).

% nat_add_max_right
tff(fact_9033_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q4)) ).

% nat_add_max_left
tff(fact_9034_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ).

% nat_mult_max_left
tff(fact_9035_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_max_right
tff(fact_9036_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9037_nat__minus__add__max,axiom,
    ! [N: 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,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_9038_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_ki,N)),M) ).

% max_Suc2
tff(fact_9039_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_kj,N)),M) ).

% max_Suc1
tff(fact_9040_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9041_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bc),Nat) ).

% pred_def
tff(fact_9042_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_kk,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9043_normalize__negative,axiom,
    ! [Q4: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q4),zero_zero_int))
     => ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3)),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_9044_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_9045_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_9046_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_9047_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9048_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9049_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9050_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_9051_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9052_divide__rat__def,axiom,
    ! [Q4: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q4),R2) = aa_rat_rat(times_times_rat(Q4),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9053_diff__rat__def,axiom,
    ! [Q4: rat,R2: rat] : aa_rat_rat(minus_minus_rat(Q4),R2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q4),aa_rat_rat(uminus_uminus_rat,R2)) ).

% diff_rat_def
tff(fact_9054_quotient__of__div,axiom,
    ! [R2: rat,N: int,D: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(N),D) )
     => ( R2 = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,N)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9055_rat__plus__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P3),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_km(Q4),quotient_of(P3)) ).

% rat_plus_code
tff(fact_9056_rat__minus__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P3),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_ko(Q4),quotient_of(P3)) ).

% rat_minus_code
tff(fact_9057_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q4: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% quotient_of_denom_pos
tff(fact_9058_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_kp,quotient_of(P3)) ).

% rat_uminus_code
tff(fact_9059_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q4: int] :
      ( ( normalize(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% normalize_denom_pos
tff(fact_9060_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P3: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9061_rat__less__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q4))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_kr(Q4)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_9062_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa3: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa3) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa3),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa3)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa3),X))
         => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa3),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_9063_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(product_Pair_int_int(A),one_one_int) ).

% quotient_of_int
tff(fact_9064_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(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_9065_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ? [R4: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R4),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S3))))
          & ! [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N6),aa_set_nat_nat(finite_card_nat,S3)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R4,N6)),S3)) ) ) ) ).

% finite_enumerate
tff(fact_9066_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_ks(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_9067_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9068_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_9069_drop__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9070_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9071_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(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_9072_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9073_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),aa_int_int(bit_se545348938243370406it_int(N),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9074_Rat_Oof__int__def,axiom,
    of_int = ring_1_of_int_rat ).

% Rat.of_int_def
tff(fact_9075_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(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9076_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9077_Frct__code__post_I9_J,axiom,
    ! [Q4: product_prod_int_int] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,frct(Q4))) = frct(Q4) ).

% Frct_code_post(9)
tff(fact_9078_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(N),K)) ).

% drop_bit_nat_eq
tff(fact_9079_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_9080_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_9081_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),aa_int_int(uminus_uminus_int,B))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B))) ).

% Frct_code_post(8)
tff(fact_9082_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,A)),B)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B))) ).

% Frct_code_post(7)
tff(fact_9083_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9084_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_9085_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_kt(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_ku(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_9086_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( ( L = zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
          & ( ( L != zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_kv(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_9087_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(aa_num_int(neg_numeral_sub_int(N),one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9088_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] : aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K),aa_num_int(neg_numeral_sub_int(N),one))) ).

% xor_minus_numerals(2)
tff(fact_9089_sub__BitM__One__eq,axiom,
    ! [N: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(N),one)) ).

% sub_BitM_One_eq
tff(fact_9090_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9091_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_ag,aTP_Lamm_af) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9092_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_9093_card_Ocomp__fun__commute__on,axiom,
    comp_nat_nat_nat(suc,suc) = comp_nat_nat_nat(suc,suc) ).

% card.comp_fun_commute_on
tff(fact_9094_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9095_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Code_integer_nat(code_nat_of_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(nat2,X) ).

% nat_of_integer.abs_eq
tff(fact_9096_nat__of__integer__code__post_I2_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9097_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9098_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_9099_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K) = produc1555791787009142072er_nat(aTP_Lamm_kw,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_9100_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_int(code_int_of_integer,K) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( aa_Code_integer_int(code_int_of_integer,K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( aa_Code_integer_int(code_int_of_integer,K) = produc1553301316500091796er_int(aTP_Lamm_kx,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9101_int__of__integer__of__nat,axiom,
    ! [N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% int_of_integer_of_nat
tff(fact_9102_of__int__integer__of,axiom,
    ! [K: code_integer] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_Code_integer_int(code_int_of_integer,K)) = K ).

% of_int_integer_of
tff(fact_9103_int__of__integer__of__int,axiom,
    ! [K: int] : aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(ring_18347121197199848620nteger,K)) = K ).

% int_of_integer_of_int
tff(fact_9104_zero__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9105_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Xa3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa3)) ).

% plus_integer.rep_eq
tff(fact_9106_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,X)) ).

% uminus_integer.rep_eq
tff(fact_9107_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9108_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Xa3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa3)) ).

% minus_integer.rep_eq
tff(fact_9109_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa3))) ) ).

% less_integer.rep_eq
tff(fact_9110_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L))) ) ).

% integer_less_iff
tff(fact_9111_nat__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_nat(code_nat_of_integer,X) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% nat_of_integer.rep_eq
tff(fact_9112_times__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa3)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_kz),Xa3),X)) ).

% times_int.abs_eq
tff(fact_9113_Gcd__remove0__nat,axiom,
    ! [M10: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M10))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M10) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M10,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9114_Gcd__nat__eq__one,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(one_one_nat),N5))
     => ( aa_set_nat_nat(gcd_Gcd_nat,N5) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9115_int_Oabs__induct,axiom,
    ! [P: fun_int_bool,X: int] :
      ( ! [Y3: product_prod_nat_nat] : pp(aa_int_bool(P,aa_Pro4240683299535347697at_int(abs_Integ,Y3)))
     => pp(aa_int_bool(P,X)) ) ).

% int.abs_induct
tff(fact_9116_Gcd__greatest__nat,axiom,
    ! [A2: set_nat,A: nat] :
      ( ! [B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(B3),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B3)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_set_nat_nat(gcd_Gcd_nat,A2))) ) ).

% Gcd_greatest_nat
tff(fact_9117_Gcd__dvd__nat,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(gcd_Gcd_nat,A2)),A)) ) ).

% Gcd_dvd_nat
tff(fact_9118_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)) ).

% eq_Abs_Integ
tff(fact_9119_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_9120_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9121_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_9122_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_la),X)) ).

% uminus_int.abs_eq
tff(fact_9123_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9124_less__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa3)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_lc),Xa3),X)) ) ).

% less_int.abs_eq
tff(fact_9125_less__eq__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa3)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_le),Xa3),X)) ) ).

% less_eq_int.abs_eq
tff(fact_9126_plus__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa3)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_lg),Xa3),X)) ).

% plus_int.abs_eq
tff(fact_9127_minus__int_Oabs__eq,axiom,
    ! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa3)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_li),Xa3),X)) ).

% minus_int.abs_eq
tff(fact_9128_csqrt_Ocode,axiom,
    ! [Z: complex] : aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9129_abs__Gcd__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Gcd_int(K5)) = gcd_Gcd_int(K5) ).

% abs_Gcd_eq
tff(fact_9130_complex__Im__of__int,axiom,
    ! [Z: int] : aa_complex_real(im,aa_int_complex(ring_17405671764205052669omplex,Z)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_9131_complex__Re__of__int,axiom,
    ! [Z: int] : aa_complex_real(re,aa_int_complex(ring_17405671764205052669omplex,Z)) = aa_int_real(ring_1_of_int_real,Z) ).

% complex_Re_of_int
tff(fact_9132_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9133_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,X)),zero_zero_real))
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).

% csqrt_minus
tff(fact_9134_Gcd__greatest__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B3: int] :
          ( member_int(B3,A2)
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B3)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),gcd_Gcd_int(A2))) ) ).

% Gcd_greatest_int
tff(fact_9135_Gcd__dvd__int,axiom,
    ! [A: int,A2: set_int] :
      ( member_int(A,A2)
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,gcd_Gcd_int(A2)),A)) ) ).

% Gcd_dvd_int
tff(fact_9136_uminus__complex_Ocode,axiom,
    ! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,X) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,aa_complex_real(re,X))),aa_real_real(uminus_uminus_real,aa_complex_real(im,X))) ).

% uminus_complex.code
tff(fact_9137_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( member_complex(Z,ring_1_Ints_complex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I3: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I3) ) ) ).

% complex_is_Int_iff
tff(fact_9138_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9139_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9140_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9141_uminus__complex_Osimps_I2_J,axiom,
    ! [X: complex] : aa_complex_real(im,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,X)) ).

% uminus_complex.simps(2)
tff(fact_9142_uminus__complex_Osimps_I1_J,axiom,
    ! [X: complex] : aa_complex_real(re,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_real_real(uminus_uminus_real,aa_complex_real(re,X)) ).

% uminus_complex.simps(1)
tff(fact_9143_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5))) ).

% Gcd_int_greater_eq_0
tff(fact_9144_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(divide_divide_real(aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9145_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9146_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9147_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa_complex_complex(invers8013647133539491842omplex,X) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9148_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real))
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9149_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9150_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9151_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa3))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_le),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa3))) ) ).

% less_eq_int.rep_eq
tff(fact_9152_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).

% nat.rep_eq
tff(fact_9153_less__int_Orep__eq,axiom,
    ! [X: int,Xa3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa3))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_lc),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa3))) ) ).

% less_int.rep_eq
tff(fact_9154_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_la)) ).

% uminus_int_def
tff(fact_9155_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_lj) ).

% prod_encode_def
tff(fact_9156_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_9157_prod__encode__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,X) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Y) )
    <=> ( X = Y ) ) ).

% prod_encode_eq
tff(fact_9158_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q4)) = Q4 ).

% num_of_nat_numeral_eq
tff(fact_9159_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9160_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_1
tff(fact_9161_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_2
tff(fact_9162_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_9163_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_9164_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_9165_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9166_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_9167_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_kz)) ).

% times_int_def
tff(fact_9168_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_li)) ).

% minus_int_def
tff(fact_9169_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_lg)) ).

% plus_int_def
tff(fact_9170_num__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Code_integer_num(code_num_of_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_num(num_of_nat,aa_int_nat(nat2,X)) ).

% num_of_integer.abs_eq
tff(fact_9171_num__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_num(code_num_of_integer,X) = aa_nat_num(num_of_nat,aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X))) ).

% num_of_integer.rep_eq
tff(fact_9172_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( aa_Code_integer_num(code_num_of_integer,K) = one ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( aa_Code_integer_num(code_num_of_integer,K) = produc7336495610019696514er_num(aTP_Lamm_lk,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_9173_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : aa_num_num(pow(X),aa_num_num(bit1,Y)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(X),Y))),X) ).

% pow.simps(3)
tff(fact_9174_sqr_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,N))) ).

% sqr.simps(2)
tff(fact_9175_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9176_sqr__conv__mult,axiom,
    ! [X: num] : aa_num_num(sqr,X) = aa_num_num(times_times_num(X),X) ).

% sqr_conv_mult
tff(fact_9177_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : aa_num_num(pow(X),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(pow(X),Y)) ).

% pow.simps(2)
tff(fact_9178_sqr_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(sqr,N)),N))) ).

% sqr.simps(3)
tff(fact_9179_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9180_bij__betw__Suc,axiom,
    ! [M10: set_nat,N5: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M10),N5))
    <=> ( image_nat_nat(suc,M10) = N5 ) ) ).

% bij_betw_Suc
tff(fact_9181_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9182_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9183_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).

% zero_notin_Suc_image
tff(fact_9184_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_9185_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9186_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9187_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9188_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9189_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_9190_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int)))) ) ).

% rat_floor_lemma
tff(fact_9191_minus__rat__cancel,axiom,
    ! [A: int,B: int] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ).

% minus_rat_cancel
tff(fact_9192_inverse__rat,axiom,
    ! [A: int,B: int] : aa_rat_rat(inverse_inverse_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int_rat(aa_int_fun_int_rat(fract,B),A) ).

% inverse_rat
tff(fact_9193_Gcd__abs__eq,axiom,
    ! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).

% Gcd_abs_eq
tff(fact_9194_minus__rat,axiom,
    ! [A: int,B: int] : aa_rat_rat(uminus_uminus_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_rat
tff(fact_9195_Gcd__int__eq,axiom,
    ! [N5: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N5)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N5)) ).

% Gcd_int_eq
tff(fact_9196_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_lm,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9197_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% less_rat
tff(fact_9198_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_int_rat(fract,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,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% add_rat
tff(fact_9199_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% le_rat
tff(fact_9200_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(minus_minus_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,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% diff_rat
tff(fact_9201_sgn__rat,axiom,
    ! [A: int,B: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B))) ).

% sgn_rat
tff(fact_9202_eq__rat_I3_J,axiom,
    ! [A: int,C: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),A) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),C) ).

% eq_rat(3)
tff(fact_9203_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(aa_int_fun_int_rat(fract,A),zero_zero_int) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_9204_Rat__induct__pos,axiom,
    ! [P: fun_rat_bool,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3))) )
     => pp(aa_rat_bool(P,Q4)) ) ).

% Rat_induct_pos
tff(fact_9205_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(aa_int_fun_int_rat(fract,A),B) = aa_int_rat(aa_int_fun_int_rat(fract,C),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9206_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,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_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% mult_rat_cancel
tff(fact_9207_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) = aa_nat_rat(semiri681578069525770553at_rat,K) ).

% Fract_of_nat_eq
tff(fact_9208_rat__number__collapse_I1_J,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),K) = zero_zero_rat ).

% rat_number_collapse(1)
tff(fact_9209_rat__number__collapse_I6_J,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),zero_zero_int) = zero_zero_rat ).

% rat_number_collapse(6)
tff(fact_9210_One__rat__def,axiom,
    one_one_rat = aa_int_rat(aa_int_fun_int_rat(fract,one_one_int),one_one_int) ).

% One_rat_def
tff(fact_9211_Fract__of__int__eq,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),one_one_int) = aa_int_rat(ring_1_of_int_rat,K) ).

% Fract_of_int_eq
tff(fact_9212_Fract__of__int__quotient,axiom,
    ! [K: int,L: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),L) = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L)) ).

% Fract_of_int_quotient
tff(fact_9213_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9214_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_9215_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa_num_rat(numeral_numeral_rat,K) = aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% rat_number_expand(3)
tff(fact_9216_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,W)),one_one_int) = aa_num_rat(numeral_numeral_rat,W) ).

% rat_number_collapse(3)
tff(fact_9217_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_9218_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% zero_less_Fract_iff
tff(fact_9219_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Fract_less_zero_iff
tff(fact_9220_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% Fract_less_one_iff
tff(fact_9221_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_9222_rat__number__collapse_I5_J,axiom,
    aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% rat_number_collapse(5)
tff(fact_9223_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)),N) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,M),N)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9224_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S3))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9225_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_ln(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_9226_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_Fract_iff
tff(fact_9227_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% Fract_le_zero_iff
tff(fact_9228_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% Fract_le_one_iff
tff(fact_9229_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% one_le_Fract_iff
tff(fact_9230_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) ).

% rat_number_collapse(4)
tff(fact_9231_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% rat_number_expand(5)
tff(fact_9232_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_9233_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_9234_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_lo(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9235_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% positive_rat
tff(fact_9236_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9237_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_9238_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9239_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_lc)) ).

% less_int_def
tff(fact_9240_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_le)) ).

% less_eq_int_def
tff(fact_9241_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9242_Rat_Opositive__minus,axiom,
    ! [X: rat] :
      ( ~ pp(aa_rat_bool(positive,X))
     => ( ( X != zero_zero_rat )
       => pp(aa_rat_bool(positive,aa_rat_rat(uminus_uminus_rat,X))) ) ) ).

% Rat.positive_minus
tff(fact_9243_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_9244_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).

% and_minus_numerals(7)
tff(fact_9245_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).

% and_minus_numerals(3)
tff(fact_9246_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).

% and_minus_numerals(4)
tff(fact_9247_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_9248_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_lp,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_9249_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).

% and_minus_numerals(8)
tff(fact_9250_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9251_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_9252_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_9253_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_9254_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9255_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9256_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_9257_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_9258_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9259_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9260_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_9261_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9262_min__diff,axiom,
    ! [M: nat,I: nat,N: 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,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_9263_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ).

% nat_mult_min_left
tff(fact_9264_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_min_right
tff(fact_9265_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9266_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9267_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] : aa_int_int(bit_concat_bit(M,aa_int_int(bit_concat_bit(N,K),L)),R2) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9268_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_concat_bit(N,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9269_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_lq,N)),M) ).

% min_Suc2
tff(fact_9270_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_lr,N)),M) ).

% min_Suc1
tff(fact_9271_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9272_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9273_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9274_vanishes__mult__bounded,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( ? [A6: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A6))
          & ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N2))),A6)) )
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ls,X7),Y6))) ) ) ).

% vanishes_mult_bounded
tff(fact_9275_vanishes__const,axiom,
    ! [C: rat] :
      ( pp(vanishes(aTP_Lamm_lt(C)))
    <=> ( C = zero_zero_rat ) ) ).

% vanishes_const
tff(fact_9276_vanishes__minus,axiom,
    ! [X7: fun_nat_rat] :
      ( pp(vanishes(X7))
     => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_lu,X7))) ) ).

% vanishes_minus
tff(fact_9277_vanishes__add,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( pp(vanishes(X7))
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_lv,X7),Y6))) ) ) ).

% vanishes_add
tff(fact_9278_vanishes__diff,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( pp(vanishes(X7))
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_lw(X7),Y6))) ) ) ).

% vanishes_diff
tff(fact_9279_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_lx(M),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ly(M),N5)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_lz(M),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9280_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_lx(M),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),M)),one_one_nat)),N5) ).

% card_length_sum_list
tff(fact_9281_vanishes__def,axiom,
    ! [X7: fun_nat_rat] :
      ( pp(vanishes(X7))
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N4))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9282_vanishesI,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [R4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R4))
         => ? [K4: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N2))),R4)) ) )
     => pp(vanishes(X7)) ) ).

% vanishesI
tff(fact_9283_vanishesD,axiom,
    ! [X7: fun_nat_rat,R2: rat] :
      ( pp(vanishes(X7))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N6))),R2)) ) ) ) ).

% vanishesD
tff(fact_9284_rcis__inverse,axiom,
    ! [R2: real,A: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(rcis(R2),A)) = aa_real_complex(rcis(aa_real_real(divide_divide_real(one_one_real),R2)),aa_real_real(uminus_uminus_real,A)) ).

% rcis_inverse
tff(fact_9285_cis__rcis__eq,axiom,
    ! [A: real] : aa_real_complex(cis,A) = aa_real_complex(rcis(one_one_real),A) ).

% cis_rcis_eq
tff(fact_9286_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa3))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa3 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( ( Deg2 = Xa3 )
                & ! [X3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9287_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa3))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa3 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ~ ( ( Deg2 = Xa3 )
                  & ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                  & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9288_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList2,Summary)),Deg4))
    <=> ( ( Deg = Deg4 )
        & ! [X5: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),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(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg),TreeList2),Summary)),Mima2) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9289_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa3))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa3 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa3 )
                    & ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9290_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa3))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa3 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa3)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa3 )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg2),TreeList),Summary2)),Mima) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa3)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9291_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa3))
               => ( Xa3 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa3))
                 => ~ ( ( Deg2 = Xa3 )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9292_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa3))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa3))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa3))
               => ( Xa3 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa3))
                 => ( ( Deg2 = Xa3 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9293_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9294_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9295_length__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_nat(size_size_list_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9296_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9297_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9298_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9299_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_9300_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_9301_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9302_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_9303_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9304_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_9305_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_9306_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_9307_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_mc,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9308_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9309_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_md) ).

% Rats_eq_int_div_nat
tff(fact_9310_Rats__abs__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(abs_abs_real,X),field_5140801741446780682s_real)
    <=> member_real(X,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_9311_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X3)) ) ).

% Rats_no_top_le
tff(fact_9312_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X3: real] :
          ( member_real(X3,field_5140801741446780682s_real)
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9313_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X)) ) ).

% Rats_no_bot_less
tff(fact_9314_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_me) ).

% Rats_eq_int_div_int
tff(fact_9315_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_9316_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] : X != cons_nat(X3,Xs2) ) ).

% list_encode.cases
tff(fact_9317_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q4) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9318_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9319_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9320_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9321_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9322_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9323_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X3,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9324_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9325_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9326_list__encode__eq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = aa_list_nat_nat(nat_list_encode,Y) )
    <=> ( X = Y ) ) ).

% list_encode_eq
tff(fact_9327_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9328_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9329_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9330_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9331_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_9332_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,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_9333_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).

% drop_upt
tff(fact_9334_upto_Opelims,axiom,
    ! [X: int,Xa3: int,Y: list_int] :
      ( ( upto(X,Xa3) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa3))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa3))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa3)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa3))
               => ( Y = nil_int ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa3)) ) ) ) ).

% upto.pelims
tff(fact_9335_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(I),J))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_9336_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_9337_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_9338_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_9339_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9340_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9341_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9342_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_9343_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_9344_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_9345_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9346_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).

% sorted_wrt_upto
tff(fact_9347_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9348_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9349_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9350_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9351_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9352_upto_Oelims,axiom,
    ! [X: int,Xa3: int,Y: list_int] :
      ( ( upto(X,Xa3) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa3))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa3)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa3))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_9353_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_9354_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9355_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9356_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9357_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( aa_list_nat_nat(hd_nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9358_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_9359_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X3,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9360_Code__Target__Nat_ONat_Oabs__eq,axiom,
    ! [X: int] : aa_Code_integer_nat(code_Target_Nat,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(nat2,X) ).

% Code_Target_Nat.Nat.abs_eq
tff(fact_9361_Code__Target__Nat_ONat_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_nat(code_Target_Nat,X) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% Code_Target_Nat.Nat.rep_eq
tff(fact_9362_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mf,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9363_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9364_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_9365_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mg,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9366_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_9367_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9368_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9369_infinite__UNIV__int,axiom,
    ~ finite_finite_int(top_top_set_int) ).

% infinite_UNIV_int
tff(fact_9370_bij__list__encode,axiom,
    pp(aa_set_nat_bool(bij_be8532844293280997160at_nat(nat_list_encode,top_top_set_list_nat),top_top_set_nat)) ).

% bij_list_encode
tff(fact_9371_surj__list__encode,axiom,
    image_list_nat_nat(nat_list_encode,top_top_set_list_nat) = top_top_set_nat ).

% surj_list_encode
tff(fact_9372_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_9373_bij__prod__encode,axiom,
    pp(aa_set_nat_bool(bij_be5333170631980326235at_nat(nat_prod_encode,top_to4669805908274784177at_nat),top_top_set_nat)) ).

% bij_prod_encode
tff(fact_9374_int__in__range__abs,axiom,
    ! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9375_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9376_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_mh,N)),X) ) ) ) ).

% root_def
tff(fact_9377_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D6: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D6 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D6 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
               => ( D6 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D6,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9378_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9379_Inf__real__def,axiom,
    ! [X7: set_real] : comple4887499456419720421f_real(X7) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X7))) ).

% Inf_real_def
tff(fact_9380_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9381_Inf__int__def,axiom,
    ! [X7: set_int] : complete_Inf_Inf_int(X7) = aa_int_int(uminus_uminus_int,complete_Sup_Sup_int(image_int_int(uminus_uminus_int,X7))) ).

% Inf_int_def
tff(fact_9382_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(divide_divide_real(one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9383_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X: real,N: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_mi(G),N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9384_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => has_fi5821293074295781190e_real(aTP_Lamm_mj(R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9385_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(log2(B),aa_real_real(divide_divide_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9386_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_mk(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(powr_real(aa_real_real(G,X)),aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9387_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9388_DERIV__arctan,axiom,
    ! [X: 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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9389_arsinh__real__has__field__derivative,axiom,
    ! [X: 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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_9390_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D6: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D6 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D6 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D6,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9391_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => 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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9392_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9393_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_ml(F),X3)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
         => has_fi5821293074295781190e_real(aTP_Lamm_mn(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ml(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9394_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9395_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9396_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9397_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mo(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9398_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mo(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9399_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9400_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T2: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_mp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9401_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ? [T2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_mp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9402_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T2: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_mp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9403_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X != zero_zero_real )
         => ( ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mo(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9404_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,T2: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mo(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9405_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
                 => ( ( X != C )
                   => ? [T2: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T2))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T2))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_mq(Diff),C),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9406_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
             => ? [T2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),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_mr(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(B),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9407_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ? [T2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),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_mr(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(A),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9408_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M2: nat,T2: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M5: nat,T3: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_mt(N),Diff),B2),M5),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mu(Diff),M5),T3),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M5))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(T3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M5)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M5))))))),topolo2177554685111907308n_real(T3,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9409_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_mv,suminf_real(aTP_Lamm_mw(X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9410_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_mx,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_9411_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,X: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,X),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_my(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9412_surj__int__encode,axiom,
    image_int_nat(nat_int_encode,top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_9413_int__encode__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_nat(nat_int_encode,X) = aa_int_nat(nat_int_encode,Y) )
    <=> ( X = Y ) ) ).

% int_encode_eq
tff(fact_9414_bij__int__encode,axiom,
    pp(aa_set_nat_bool(bij_betw_int_nat(nat_int_encode,top_top_set_int),top_top_set_nat)) ).

% bij_int_encode
tff(fact_9415_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9416_continuous__floor,axiom,
    ! [X: real] :
      ( ~ member_real(X,ring_1_Ints_real)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),comp_int_real_real(ring_1_of_int_real,archim6058952711729229775r_real)) ) ).

% continuous_floor
tff(fact_9417_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D6: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D6,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D6 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
           => ( ! [Y3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B))
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D6),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9418_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9419_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9420_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9421_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
         => ! [N6: nat] : member_real(suminf_real(aTP_Lamm_mz(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_mz(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))),N6))),groups6591440286371151544t_real(aTP_Lamm_mz(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))),N6)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9422_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N6: nat] : member_real(suminf_real(aTP_Lamm_mz(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_mz(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))),N6)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_mz(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))),N6))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9423_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_na(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mz(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9424_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9425_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9426_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_nc,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9427_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N6)),L2))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),aa_nat_real(G,N6)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9428_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R4: real] :
        ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R4),aa_nat_real(X7,N2))) )
     => filterlim_nat_real(aTP_Lamm_nd(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9429_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_ne,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9430_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aTP_Lamm_nf(C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9431_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_ng,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9432_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_nh(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9433_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),L))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N6)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9434_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => filterlim_nat_real(power_power_real(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9435_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ni(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9436_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_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_9437_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(power_power_real(C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9438_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aTP_Lamm_nj(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9439_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_nk(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9440_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aTP_Lamm_nl(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9441_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_nm(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9442_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_mz(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9443_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => summable_real(aTP_Lamm_mz(A)) ) ) ) ).

% summable
tff(fact_9444_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_nn(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_no(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9445_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_np(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_no(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9446_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_nq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mz(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9447_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aTP_Lamm_bt(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9448_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_nq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mz(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9449_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_mz(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))),N)))),suminf_real(aTP_Lamm_mz(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9450_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_mz(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))),N6)))),L2))
              & filterlim_nat_real(aTP_Lamm_nq(A),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_mz(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))),N6)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_na(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9451_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_na(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mz(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9452_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_mz(A))),groups6591440286371151544t_real(aTP_Lamm_mz(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))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9453_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_nr(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9454_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_ns(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9455_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_ns(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9456_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9457_filterlim__real__of__int__at__top,axiom,
    filterlim_int_real(ring_1_of_int_real,at_top_real,at_top_int) ).

% filterlim_real_of_int_at_top
tff(fact_9458_filterlim__int__sequentially,axiom,
    filterlim_nat_int(semiri1314217659103216013at_int,at_top_int,at_top_nat) ).

% filterlim_int_sequentially
tff(fact_9459_filterlim__nat__sequentially,axiom,
    filterlim_int_nat(nat2,at_top_nat,at_top_int) ).

% filterlim_nat_sequentially
tff(fact_9460_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_9461_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_9462_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_nt(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_9463_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_nu(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9464_eventually__at__left__to__right,axiom,
    ! [P: fun_real_bool,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
    <=> eventually_real(aTP_Lamm_nv(P),topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,A),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,A)))) ) ).

% eventually_at_left_to_right
tff(fact_9465_eventually__at__right__to__top,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_nw(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_9466_eventually__at__top__to__right,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_nw(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_9467_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_9468_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_9469_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_9470_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_9471_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F2: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nx(N),F),at_top_real,F2) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9472_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_9473_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_9474_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_9475_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9476_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_9477_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_9478_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_9479_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F2: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nx(N),F),at_bot_real,F2) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9480_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_9481_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => bfun_nat_real(power_power_real(X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9482_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_mh,N),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9483_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X3: real] :
            ( member_real(X3,A2)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X3))) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_ny(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9484_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_9485_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_9486_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A2,artanh_real) ) ).

% continuous_on_artanh
tff(fact_9487_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X3: real] :
            ( member_real(X3,A2)
           => member_real(aa_real_real(F,X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_nz(F)) ) ) ).

% continuous_on_artanh'
tff(fact_9488_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => inj_on_real_real(log2(B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9489_surj__int__decode,axiom,
    image_nat_int(nat_int_decode,top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_9490_int__decode__inverse,axiom,
    ! [N: nat] : aa_int_nat(nat_int_encode,aa_nat_int(nat_int_decode,N)) = N ).

% int_decode_inverse
tff(fact_9491_int__encode__inverse,axiom,
    ! [X: int] : aa_nat_int(nat_int_decode,aa_int_nat(nat_int_encode,X)) = X ).

% int_encode_inverse
tff(fact_9492_inj__list__encode,axiom,
    ! [A2: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A2) ).

% inj_list_encode
tff(fact_9493_inj__prod__encode,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : inj_on2178005380612969504at_nat(nat_prod_encode,A2) ).

% inj_prod_encode
tff(fact_9494_inj__Suc,axiom,
    ! [N5: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N5)) ).

% inj_Suc
tff(fact_9495_inj__int__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on_nat_int(nat_int_decode),A2)) ).

% inj_int_decode
tff(fact_9496_inj__int__encode,axiom,
    ! [A2: set_int] : inj_on_int_nat(nat_int_encode,A2) ).

% inj_int_encode
tff(fact_9497_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll,K)),N5)) ) ).

% inj_on_diff_nat
tff(fact_9498_int__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_int(nat_int_decode,X) = aa_nat_int(nat_int_decode,Y) )
    <=> ( X = Y ) ) ).

% int_decode_eq
tff(fact_9499_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9500_bij__int__decode,axiom,
    bij_betw_nat_int(nat_int_decode,top_top_set_nat,top_top_set_int) ).

% bij_int_decode
tff(fact_9501_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)) = 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_9502_Gcd__eq__Max,axiom,
    ! [M10: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M10))
     => ( ( M10 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M10))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M10) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_oa,M10))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9503_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9504_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9505_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9506_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3)))) ) ).

% card_le_Suc_Max
tff(fact_9507_Sup__nat__def,axiom,
    ! [X7: set_nat] :
      ( ( ( X7 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X7) = zero_zero_nat ) )
      & ( ( X7 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X7) ) ) ) ).

% Sup_nat_def
tff(fact_9508_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ob(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9509_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ay,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9510_cauchy__def,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M4))
             => ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X7,M4)),aa_nat_rat(X7,N4)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9511_cauchyI,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [R4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R4))
         => ? [K4: nat] :
            ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M2))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X7,M2)),aa_nat_rat(X7,N2)))),R4)) ) ) )
     => cauchy(X7) ) ).

% cauchyI
tff(fact_9512_cauchy__add,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_lv,X7),Y6)) ) ) ).

% cauchy_add
tff(fact_9513_cauchy__mult,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ls,X7),Y6)) ) ) ).

% cauchy_mult
tff(fact_9514_cauchy__diff,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_lw(X7),Y6)) ) ) ).

% cauchy_diff
tff(fact_9515_cauchy__const,axiom,
    ! [X: rat] : cauchy(aTP_Lamm_lt(X)) ).

% cauchy_const
tff(fact_9516_cauchy__minus,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_lu,X7)) ) ).

% cauchy_minus
tff(fact_9517_cauchy__inverse,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => cauchy(aTP_Lamm_oc(X7)) ) ) ).

% cauchy_inverse
tff(fact_9518_cauchy__imp__bounded,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ? [B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
          & ! [N6: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N6))),B3)) ) ) ).

% cauchy_imp_bounded
tff(fact_9519_vanishes__diff__inverse,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ( cauchy(Y6)
         => ( ~ pp(vanishes(Y6))
           => ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_lw(X7),Y6)))
             => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_od(X7),Y6))) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_9520_cauchy__not__vanishes__cases,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
                ( ! [N6: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X7,N6)))) )
                | ! [N6: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_nat_rat(X7,N6))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9521_cauchy__not__vanishes,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
              ! [N6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N6)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9522_cauchyD,axiom,
    ! [X7: fun_nat_rat,R2: rat] :
      ( cauchy(X7)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M5))
           => ! [N6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X7,M5)),aa_nat_rat(X7,N6)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_9523_le__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X7,N4)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y6,N4)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9524_inverse__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ( pp(vanishes(X7))
         => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X7)) = zero_zero_real ) )
        & ( ~ pp(vanishes(X7))
         => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X7)) = aa_fun_nat_rat_real(real2,aTP_Lamm_oc(X7)) ) ) ) ) ).

% inverse_Real
tff(fact_9525_Real__induct,axiom,
    ! [P: fun_real_bool,X: real] :
      ( ! [X8: fun_nat_rat] :
          ( cauchy(X8)
         => pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,X8))) )
     => pp(aa_real_bool(P,X)) ) ).

% Real_induct
tff(fact_9526_zero__real__def,axiom,
    zero_zero_real = aa_fun_nat_rat_real(real2,aTP_Lamm_oe) ).

% zero_real_def
tff(fact_9527_one__real__def,axiom,
    one_one_real = aa_fun_nat_rat_real(real2,aTP_Lamm_of) ).

% one_real_def
tff(fact_9528_of__int__Real,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_og(X)) ).

% of_int_Real
tff(fact_9529_of__nat__Real,axiom,
    ! [X: nat] : aa_nat_real(semiri5074537144036343181t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_oh(X)) ).

% of_nat_Real
tff(fact_9530_minus__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X7)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_lu,X7)) ) ) ).

% minus_Real
tff(fact_9531_add__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_lv,X7),Y6)) ) ) ) ).

% add_Real
tff(fact_9532_mult__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ls,X7),Y6)) ) ) ) ).

% mult_Real
tff(fact_9533_diff__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa_real_real(minus_minus_real(aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_lw(X7),Y6)) ) ) ) ).

% diff_Real
tff(fact_9534_eq__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( ( aa_fun_nat_rat_real(real2,X7) = aa_fun_nat_rat_real(real2,Y6) )
        <=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_lw(X7),Y6))) ) ) ) ).

% eq_Real
tff(fact_9535_not__positive__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X7)))
      <=> ! [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
           => ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X7,N4)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9536_positive__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X7)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X7,N4))) ) ) ) ) ).

% positive_Real
tff(fact_9537_Real_Opositive__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(positive2,X))
     => ( pp(aa_real_bool(positive2,Y))
       => pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% Real.positive_add
tff(fact_9538_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(positive2,X))
     => ( pp(aa_real_bool(positive2,Y))
       => pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))) ) ) ).

% Real.positive_mult
tff(fact_9539_Real_Opositive__zero,axiom,
    ~ pp(aa_real_bool(positive2,zero_zero_real)) ).

% Real.positive_zero
tff(fact_9540_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(positive2,X))
     => ( ( X != zero_zero_real )
       => pp(aa_real_bool(positive2,aa_real_real(uminus_uminus_real,X))) ) ) ).

% Real.positive_minus
tff(fact_9541_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(minus_minus_real(Y),X))) ) ).

% less_real_def
tff(fact_9542_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X),N4))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9543_inverse__real_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,if_fun_nat_rat(vanishes(X),aTP_Lamm_oe,aTP_Lamm_oc(X))) ) ) ).

% inverse_real.abs_eq
tff(fact_9544_realrel__refl,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X7),X7)) ) ).

% realrel_refl
tff(fact_9545_one__real_Orsp,axiom,
    pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_of),aTP_Lamm_of)) ).

% one_real.rsp
tff(fact_9546_zero__real_Orsp,axiom,
    pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_oe),aTP_Lamm_oe)) ).

% zero_real.rsp
tff(fact_9547_real_Oabs__induct,axiom,
    ! [P: fun_real_bool,X: real] :
      ( ! [Y3: fun_nat_rat] :
          ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Y3),Y3))
         => pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,Y3))) )
     => pp(aa_real_bool(P,X)) ) ).

% real.abs_induct
tff(fact_9548_uminus__real_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_lu,X)) ) ) ).

% uminus_real.abs_eq
tff(fact_9549_plus__real_Oabs__eq,axiom,
    ! [Xa3: fun_nat_rat,X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa3),Xa3))
     => ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,Xa3)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_lv,Xa3),X)) ) ) ) ).

% plus_real.abs_eq
tff(fact_9550_times__real_Oabs__eq,axiom,
    ! [Xa3: fun_nat_rat,X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa3),Xa3))
     => ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,Xa3)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ls,Xa3),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_9551_realrelI,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_lw(X7),Y6)))
         => pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X7),Y6)) ) ) ) ).

% realrelI
tff(fact_9552_realrel__def,axiom,
    ! [X4: fun_nat_rat,Xa: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X4),Xa))
    <=> ( cauchy(X4)
        & cauchy(Xa)
        & pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_lw(X4),Xa))) ) ) ).

% realrel_def
tff(fact_9553_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X,N4))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_9554_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_oi) ).

% Real.positive_def
tff(fact_9555_inverse__real__def,axiom,
    inverse_inverse_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_oj) ).

% inverse_real_def
tff(fact_9556_uminus__real__def,axiom,
    uminus_uminus_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_lu) ).

% uminus_real_def
tff(fact_9557_times__real__def,axiom,
    times_times_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_ls) ).

% times_real_def
tff(fact_9558_plus__real__def,axiom,
    plus_plus_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_lv) ).

% plus_real_def
tff(fact_9559_Real_Opositive_Orsp,axiom,
    bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_oi,aTP_Lamm_oi) ).

% Real.positive.rsp
tff(fact_9560_cr__real__def,axiom,
    ! [X4: fun_nat_rat,Xa: real] :
      ( pp(aa_real_bool(aa_fun5258240734392065441l_bool(cr_real,X4),Xa))
    <=> ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X4),X4))
        & ( aa_fun_nat_rat_real(real2,X4) = Xa ) ) ) ).

% cr_real_def
tff(fact_9561_times__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_ls,aTP_Lamm_ls) ).

% times_real.rsp
tff(fact_9562_plus__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_lv,aTP_Lamm_lv) ).

% plus_real.rsp
tff(fact_9563_integer__of__natural_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,semiri1314217659103216013at_int,semiri1314217659103216013at_int) ).

% integer_of_natural.rsp
tff(fact_9564_uminus__integer_Orsp,axiom,
    pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),uminus_uminus_int),uminus_uminus_int)) ).

% uminus_integer.rsp
tff(fact_9565_natural__of__integer_Orsp,axiom,
    bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,nat2,nat2) ).

% natural_of_integer.rsp
tff(fact_9566_Suc_Orsp,axiom,
    pp(aa_fun_nat_nat_bool(aa_fun3185460345579526648t_bool(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc)) ).

% Suc.rsp
tff(fact_9567_less__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9568_less__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9569_dup_Orsp,axiom,
    pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_ok),aTP_Lamm_ok)) ).

% dup.rsp
tff(fact_9570_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_ol,aTP_Lamm_ol) ).

% sub.rsp
tff(fact_9571_minus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),minus_minus_int,minus_minus_int) ).

% minus_integer.rsp
tff(fact_9572_minus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),minus_minus_nat,minus_minus_nat) ).

% minus_natural.rsp
tff(fact_9573_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_9574_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_9575_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_9576_uminus__real_Orsp,axiom,
    pp(aa_fun1696419067750587730t_bool(aa_fun4901480949278917796t_bool(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_lu),aTP_Lamm_lu)) ).

% uminus_real.rsp
tff(fact_9577_inverse__real_Orsp,axiom,
    pp(aa_fun1696419067750587730t_bool(aa_fun4901480949278917796t_bool(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_oj),aTP_Lamm_oj)) ).

% inverse_real.rsp
tff(fact_9578_Real_Opositive_Otransfer,axiom,
    pp(aa_fun3352386287069783311l_bool(aa_fun7221085148252759141l_bool(bNF_re1458715823057739248l_bool(pcr_real,fequal_bool),aTP_Lamm_oi),positive2)) ).

% Real.positive.transfer
tff(fact_9579_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_kz),times_times_int) ).

% times_int.transfer
tff(fact_9580_real_Orel__eq__transfer,axiom,
    bNF_re8935275315890131625l_bool(pcr_real,bNF_re1458715823057739248l_bool(pcr_real,fequal_bool),realrel,fequal_real) ).

% real.rel_eq_transfer
tff(fact_9581_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_9582_zero__real_Otransfer,axiom,
    pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,aTP_Lamm_oe),zero_zero_real)) ).

% zero_real.transfer
tff(fact_9583_one__real_Otransfer,axiom,
    pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,aTP_Lamm_of),one_one_real)) ).

% one_real.transfer
tff(fact_9584_cr__real__eq,axiom,
    ! [X4: fun_nat_rat,Xa: real] :
      ( pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,X4),Xa))
    <=> ( cauchy(X4)
        & ( aa_fun_nat_rat_real(real2,X4) = Xa ) ) ) ).

% cr_real_eq
tff(fact_9585_uminus__real_Otransfer,axiom,
    pp(aa_fun9137951878280015556l_bool(aa_fun9159459243913713430l_bool(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_lu),uminus_uminus_real)) ).

% uminus_real.transfer
tff(fact_9586_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_9587_plus__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_lv,plus_plus_real) ).

% plus_real.transfer
tff(fact_9588_times__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_ls,times_times_real) ).

% times_real.transfer
tff(fact_9589_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_om,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9590_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_la)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_9591_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9592_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_9593_inverse__real_Otransfer,axiom,
    pp(aa_fun9137951878280015556l_bool(aa_fun9159459243913713430l_bool(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_oj),inverse_inverse_real)) ).

% inverse_real.transfer
tff(fact_9594_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_lc),ord_less_int) ).

% less_int.transfer
tff(fact_9595_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_le),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9596_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_lg),plus_plus_int) ).

% plus_int.transfer
tff(fact_9597_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_li),minus_minus_int) ).

% minus_int.transfer
tff(fact_9598_num__of__integer__def,axiom,
    code_num_of_integer = map_fu1227494855608507351um_num(code_int_of_integer,id_num,comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer_def
tff(fact_9599_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_kz),produc4528391901023682385at_nat(aTP_Lamm_kz)) ).

% times_int.rsp
tff(fact_9600_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9601_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_9602_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4240683299535347697at_int(abs_Integ,X) = aa_Pro4240683299535347697at_int(abs_Integ,Y) )
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,X),Y)) ) ).

% int.abs_eq_iff
tff(fact_9603_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_la)),produc2626176000494625587at_nat(aTP_Lamm_la))) ).

% uminus_int.rsp
tff(fact_9604_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_9605_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_9606_intrel__def,axiom,
    intrel = produc5023667602222594023t_bool(aTP_Lamm_oo) ).

% intrel_def
tff(fact_9607_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_lc),produc5023667602222594023t_bool(aTP_Lamm_lc)) ).

% less_int.rsp
tff(fact_9608_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_le),produc5023667602222594023t_bool(aTP_Lamm_le)) ).

% less_eq_int.rsp
tff(fact_9609_int_Orel__eq__transfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_9610_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_li),produc4528391901023682385at_nat(aTP_Lamm_li)) ).

% minus_int.rsp
tff(fact_9611_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_lg),produc4528391901023682385at_nat(aTP_Lamm_lg)) ).

% plus_int.rsp
tff(fact_9612_finite__le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S3),N))) ) ) ).

% finite_le_enumerate
tff(fact_9613_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9614_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K2: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K2)))
              <=> pp(aa_nat_bool(Q,K2)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9615_Least__Suc,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_nr(P))) ) ) ) ).

% Least_Suc
tff(fact_9616_Sup__real__def,axiom,
    ! [X7: set_real] : comple1385675409528146559p_real(X7) = ord_Least_real(aTP_Lamm_op(X7)) ).

% Sup_real_def
tff(fact_9617_rat__sgn__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P3)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P3)))),one_one_int) ).

% rat_sgn_code
tff(fact_9618_bezw_Oelims,axiom,
    ! [X: nat,Xa3: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa3) = Y )
     => ( ( ( Xa3 = zero_zero_nat )
         => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
        & ( ( Xa3 != zero_zero_nat )
         => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa3))))) ) ) ) ) ).

% bezw.elims
tff(fact_9619_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9620_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2)))) ).

% quotient_of_denom_pos'
tff(fact_9621_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9622_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_9623_one__mod__minus__numeral,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_9624_minus__one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_9625_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9626_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_mod_numeral
tff(fact_9627_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_9628_Divides_Oadjust__mod__def,axiom,
    ! [R2: int,L: int] :
      ( ( ( R2 = zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R2) = zero_zero_int ) )
      & ( ( R2 != zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_9629_bezw_Opelims,axiom,
    ! [X: nat,Xa3: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa3) = Y )
     => ( accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3))
       => ~ ( ( ( ( Xa3 = zero_zero_nat )
               => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
              & ( ( Xa3 != zero_zero_nat )
               => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa3,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa3))))) ) ) )
           => ~ accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3)) ) ) ) ).

% bezw.pelims
tff(fact_9630_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa3: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa3) = Y )
     => ( accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa3),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa3)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa3),X))
               => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa3),aa_nat_nat(suc,X))) ) ) )
           => ~ accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9631_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(positive,X))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_9632_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_oq) ).

% Rat.positive_def
tff(fact_9633_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( normalize(P3) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))),aa_int_int(divide_divide_int(product_snd_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( ( ( product_snd_int_int(P3) = zero_zero_int )
           => ( normalize(P3) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ) )
          & ( ( product_snd_int_int(P3) != zero_zero_int )
           => ( normalize(P3) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))),aa_int_int(divide_divide_int(product_snd_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_9634_gcd__1__int,axiom,
    ! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9635_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_neg2_int
tff(fact_9636_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_neg1_int
tff(fact_9637_abs__gcd__int,axiom,
    ! [X: int,Y: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% abs_gcd_int
tff(fact_9638_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(abs_abs_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_abs1_int
tff(fact_9639_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(abs_abs_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_abs2_int
tff(fact_9640_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9641_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_9642_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2_int
tff(fact_9643_gcd__0__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = aa_int_int(abs_abs_int,X) ).

% gcd_0_int
tff(fact_9644_gcd__0__left__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = aa_int_int(abs_abs_int,X) ).

% gcd_0_left_int
tff(fact_9645_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(abs_abs_int,X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_9646_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(abs_abs_int,Y) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_9647_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ).

% gcd_ge_0_int
tff(fact_9648_gcd__idem__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),X) = aa_int_int(abs_abs_int,X) ).

% gcd_idem_int
tff(fact_9649_gcd__red__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(modulo_modulo_int(X),Y)) ).

% gcd_red_int
tff(fact_9650_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V3),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% bezout_int
tff(fact_9651_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N)) ).

% gcd_mult_distrib_int
tff(fact_9652_gcd__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),gcd_gcd_int,gcd_gcd_int) ).

% gcd_integer.rsp
tff(fact_9653_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B)) ) ).

% gcd_le2_int
tff(fact_9654_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A)) ) ).

% gcd_le1_int
tff(fact_9655_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_bool] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_9656_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
        & ! [E3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),A))
              & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),B)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),D)) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9657_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(modulo_modulo_int(X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9658_gcd__code__int,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,L),zero_zero_int),K),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9659_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_9660_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_or(N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9661_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_os,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9662_inverse__rat_Otransfer,axiom,
    pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ot),inverse_inverse_rat)) ).

% inverse_rat.transfer
tff(fact_9663_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9664_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_9665_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9666_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_9667_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_9668_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_9669_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9670_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_9671_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_9672_gcd__nat_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.bounded_iff
tff(fact_9673_gcd__nat_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb2
tff(fact_9674_gcd__nat_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb1
tff(fact_9675_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9676_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9677_gcd__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) ).

% gcd_int_int_eq
tff(fact_9678_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_9679_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% gcd_nat_abs_left_eq
tff(fact_9680_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).

% gcd_le1_nat
tff(fact_9681_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ) ).

% gcd_le2_nat
tff(fact_9682_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff1_nat
tff(fact_9683_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff2_nat
tff(fact_9684_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,K),L) = abs_abs_Code_integer(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,L),zero_z3403309356797280102nteger),K,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,L),modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9685_gcd__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,X),Xa3)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa3)) ).

% gcd_integer.rep_eq
tff(fact_9686_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ).

% gcd_red_nat
tff(fact_9687_gcd__unique__nat,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
        & ! [E3: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E3),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E3),B)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E3),D)) ) )
    <=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_unique_nat
tff(fact_9688_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
        & ( B != C ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_9689_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
        & ( A != C ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_9690_gcd__nat_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
        & ( A != B ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_9691_gcd__nat_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
        & ( A != aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) )
     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
            & ( A != B ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
              & ( A != C ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_9692_gcd__nat_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)) ) ).

% gcd_nat.coboundedI2
tff(fact_9693_gcd__nat_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)) ) ).

% gcd_nat.coboundedI1
tff(fact_9694_gcd__nat_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb_iff2
tff(fact_9695_gcd__nat_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb_iff1
tff(fact_9696_gcd__nat_Ocobounded2,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ).

% gcd_nat.cobounded2
tff(fact_9697_gcd__nat_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ).

% gcd_nat.cobounded1
tff(fact_9698_gcd__nat_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.order_iff
tff(fact_9699_gcd__nat_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))) ) ) ).

% gcd_nat.boundedI
tff(fact_9700_gcd__nat_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.boundedE
tff(fact_9701_gcd__nat_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
        & ( B != A ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb4
tff(fact_9702_gcd__nat_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb3
tff(fact_9703_gcd__nat_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% gcd_nat.orderI
tff(fact_9704_gcd__nat_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.orderE
tff(fact_9705_gcd__nat_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D))) ) ) ).

% gcd_nat.mono
tff(fact_9706_gcd__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa3)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa3),X)) ).

% gcd_integer.abs_eq
tff(fact_9707_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% gcd_mult_distrib_nat
tff(fact_9708_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa3) = Y )
     => ( ( ( Xa3 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa3 != zero_zero_nat )
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_9709_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_9710_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9711_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(A3),A2))
         => ( pp(aa_set_nat_bool(member_nat(B3),A2))
           => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A2)) ) )
     => ( ( A2 != bot_bot_set_nat )
       => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A2)),A2)) ) ) ).

% Gcd_in
tff(fact_9712_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9713_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9714_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9715_gcd__int__def,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).

% gcd_int_def
tff(fact_9716_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_ou) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9717_one__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),one_one_rat)) ).

% one_rat.transfer
tff(fact_9718_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ov(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9719_zero__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),zero_zero_rat)) ).

% zero_rat.transfer
tff(fact_9720_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_ow,fract) ).

% Fract.transfer
tff(fact_9721_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9722_uminus__rat_Otransfer,axiom,
    pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ox),uminus_uminus_rat)) ).

% uminus_rat.transfer
tff(fact_9723_Rat_Opositive_Otransfer,axiom,
    bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_oq,positive) ).

% Rat.positive.transfer
tff(fact_9724_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa3) = Y )
     => ( accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3))
       => ~ ( ( ( ( Xa3 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa3 != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa3)) ) ) )
           => ~ accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa3)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9725_gcd__idem__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).

% gcd_idem_nat
tff(fact_9726_gcd__nat_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.right_idem
tff(fact_9727_gcd__nat_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.left_idem
tff(fact_9728_gcd__nat_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),A) = A ).

% gcd_nat.idem
tff(fact_9729_of__rat__Real,axiom,
    ! [X: rat] : aa_rat_real(field_7254667332652039916t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_lt(X)) ).

% of_rat_Real
tff(fact_9730_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Q2: rat] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,Q2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q2)),Y)) ) ) ).

% of_rat_dense
tff(fact_9731_less__RealD,axiom,
    ! [Y6: fun_nat_rat,X: real] :
      ( cauchy(Y6)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_fun_nat_rat_real(real2,Y6)))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N2)))) ) ) ).

% less_RealD
tff(fact_9732_le__RealI,axiom,
    ! [Y6: fun_nat_rat,X: real] :
      ( cauchy(Y6)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N2))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_fun_nat_rat_real(real2,Y6))) ) ) ).

% le_RealI
tff(fact_9733_Real__leI,axiom,
    ! [X7: fun_nat_rat,Y: real] :
      ( cauchy(X7)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X7,N2))),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X7)),Y)) ) ) ).

% Real_leI
tff(fact_9734_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_os) ).

% plus_rat_def
tff(fact_9735_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ot) ).

% inverse_rat_def
tff(fact_9736_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_9737_Fract_Oabs__eq,axiom,
    ! [Xa3: int,X: int] : aa_int_rat(aa_int_fun_int_rat(fract,Xa3),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Xa3),X))) ).

% Fract.abs_eq
tff(fact_9738_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_9739_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ox) ).

% uminus_rat_def
tff(fact_9740_plus__rat_Oabs__eq,axiom,
    ! [Xa3: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa3),Xa3))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa3)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,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,product_fst_int_int(Xa3)),product_snd_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa3)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa3)),product_snd_int_int(X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9741_ratrel__iff,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),Y))
    <=> ( ( product_snd_int_int(X) != zero_zero_int )
        & ( product_snd_int_int(Y) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Y)),product_snd_int_int(X)) ) ) ) ).

% ratrel_iff
tff(fact_9742_one__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int))) ).

% one_rat.rsp
tff(fact_9743_zero__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int))) ).

% zero_rat.rsp
tff(fact_9744_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_ow,aTP_Lamm_ow) ).

% Fract.rsp
tff(fact_9745_ratrel__def,axiom,
    ! [X4: product_prod_int_int,Xa: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X4),Xa))
    <=> ( ( product_snd_int_int(X4) != zero_zero_int )
        & ( product_snd_int_int(Xa) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X4)),product_snd_int_int(Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa)),product_snd_int_int(X4)) ) ) ) ).

% ratrel_def
tff(fact_9746_uminus__rat_Orsp,axiom,
    pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ox),aTP_Lamm_ox)) ).

% uminus_rat.rsp
tff(fact_9747_Rat_Opositive_Orsp,axiom,
    bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_oq,aTP_Lamm_oq) ).

% Rat.positive.rsp
tff(fact_9748_inverse__rat_Orsp,axiom,
    pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ot),aTP_Lamm_ot)) ).

% inverse_rat.rsp
tff(fact_9749_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_os,aTP_Lamm_os) ).

% plus_rat.rsp
tff(fact_9750_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,product_fst_int_int(X))),product_snd_int_int(X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_9751_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( pp(aa_rat_bool(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_9752_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(X)),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(X)),product_fst_int_int(X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_9753_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_oy(X)) ).

% quotient_of_def
tff(fact_9754_normalize__stable,axiom,
    ! [Q4: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P3),Q4))
       => ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4) ) ) ) ).

% normalize_stable
tff(fact_9755_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,B),C))
       => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,B)),aa_int_int(abs_abs_int,D)) )
        <=> ( ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) )
            & ( aa_int_int(abs_abs_int,C) = aa_int_int(abs_abs_int,D) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_9756_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3)) ) ) ).

% Rat_cases
tff(fact_9757_Rat__induct,axiom,
    ! [P: fun_rat_bool,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3))
           => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3))) ) )
     => pp(aa_rat_bool(P,Q4)) ) ).

% Rat_induct
tff(fact_9758_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),B))
         => ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_9759_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ( ( A3 != zero_zero_int )
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3)) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9760_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(X3)),product_snd_int_int(X3)) )
      & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(X3)))
      & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(X3)),product_snd_int_int(X3)))
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Y4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4))) )
         => ( Y4 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_9761_coprime__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M),N)) ) ).

% coprime_int_iff
tff(fact_9762_coprime__nat__abs__left__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% coprime_nat_abs_left_iff
tff(fact_9763_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).

% coprime_nat_abs_right_iff
tff(fact_9764_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),D))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,B),C))
       => ( ( 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) )
        <=> ( ( A = B )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_9765_coprime__Suc__left__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,N)),N)) ).

% coprime_Suc_left_nat
tff(fact_9766_coprime__Suc__right__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,N))) ).

% coprime_Suc_right_nat
tff(fact_9767_coprime__Suc__0__left,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).

% coprime_Suc_0_left
tff(fact_9768_coprime__Suc__0__right,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_9769_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),B))
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_9770_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_9771_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_9772_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( member_real(X,field_5140801741446780682s_real)
     => ~ ! [M2: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M2),N2)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9773_product__atMost__eq__Un,axiom,
    ! [A2: set_nat,M: nat] : produc457027306803732586at_nat(A2,aTP_Lamm_oz(M)) = sup_su6327502436637775413at_nat(produc457027306803732586at_nat(A2,aTP_Lamm_pa(M)),produc457027306803732586at_nat(A2,aTP_Lamm_pb(M))) ).

% product_atMost_eq_Un
tff(fact_9774_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_pc(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_pa(M)) ).

% pairs_le_eq_Sigma
tff(fact_9775_Code__Target__Nat_ONat__def,axiom,
    code_Target_Nat = map_fu6539832666145259331at_nat(code_int_of_integer,id_nat,nat2) ).

% Code_Target_Nat.Nat_def
tff(fact_9776_nat__of__integer__def,axiom,
    code_nat_of_integer = map_fu6539832666145259331at_nat(code_int_of_integer,id_nat,nat2) ).

% nat_of_integer_def
tff(fact_9777_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A2,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_9778_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( aa_list_nat_nat(last_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9779_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9780_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_9781_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9782_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M10: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_eh(F),M10),at_top_nat)
     => ( ! [M2: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),M2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,M2))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_9783_convergent__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo7531315842566124627t_real(power_power_real(X)) ) ) ).

% convergent_realpow
tff(fact_9784_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M10: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_eh(F),M10),at_top_nat)
     => ( ! [M2: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M10),M2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,M2)),aa_nat_real(F,N2))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_9785_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_9786_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ).

% pair_less_iff1
tff(fact_9787_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_9788_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9789_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_ou),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_9790_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_ag,aTP_Lamm_af),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_9791_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_9792_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,N),A2)) = insert_nat(N,vimage_nat_nat(suc,A2)) ).

% vimage_Suc_insert_Suc
tff(fact_9793_finite__vimage__Suc__iff,axiom,
    ! [F2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F2)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,F2)) ) ).

% finite_vimage_Suc_iff
tff(fact_9794_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_9795_euclidean__size__int__def,axiom,
    euclid4774559944035922753ze_int = comp_int_nat_int(nat2,abs_abs_int) ).

% euclidean_size_int_def
tff(fact_9796_set__decode__div__2,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,X)) ).

% set_decode_div_2
tff(fact_9797_abs__division__segment,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,aa_int_int(euclid3395696857347342551nt_int,K)) = one_one_int ).

% abs_division_segment
tff(fact_9798_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(sgn_sgn_int,K) ) ) ).

% division_segment_eq_sgn
tff(fact_9799_division__segment__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,N) = one_one_nat ).

% division_segment_nat_def
tff(fact_9800_division__segment__int__def,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = one_one_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% division_segment_int_def
tff(fact_9801_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_pd(N),K))) ).

% binomial_def
tff(fact_9802_transp__realrel,axiom,
    transp_fun_nat_rat(realrel) ).

% transp_realrel
tff(fact_9803_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_pe)) ).

% pred_nat_def
tff(fact_9804_less__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ord_le72135733267957522d_enat(M,N)
    <=> extend8126323780198025762t_bool(aTP_Lamm_pf(N),fFalse,M) ) ).

% less_enat_def
tff(fact_9805_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_pg,Qr) ).

% Divides.adjust_div_def
tff(fact_9806_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_9807_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) ).

% prod_decode_triangle_add
tff(fact_9808_prod__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode,X) = aa_nat7603221163087113075at_nat(nat_prod_decode,Y) )
    <=> ( X = Y ) ) ).

% prod_decode_eq
tff(fact_9809_prod__decode__inverse,axiom,
    ! [N: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) = N ).

% prod_decode_inverse
tff(fact_9810_prod__encode__inverse,axiom,
    ! [X: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_Pro4243173770044397973at_nat(nat_prod_encode,X)) = X ).

% prod_encode_inverse
tff(fact_9811_inj__prod__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on5538052773655684606at_nat(nat_prod_decode),A2)) ).

% inj_prod_decode
tff(fact_9812_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_9813_surj__prod__decode,axiom,
    image_5846123807819985514at_nat(nat_prod_decode,top_top_set_nat) = top_to4669805908274784177at_nat ).

% surj_prod_decode
tff(fact_9814_bij__prod__decode,axiom,
    bij_be8693218025023041337at_nat(nat_prod_decode,top_top_set_nat,top_to4669805908274784177at_nat) ).

% bij_prod_decode
tff(fact_9815_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)))
             => ( ! [X4: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N2) )
                   => pp(aa_nat_bool(P,Y4)) )
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_9816_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( X = aa_nat_nat(suc,N2) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_ph,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_9817_list__decode__inverse,axiom,
    ! [N: nat] : aa_list_nat_nat(nat_list_encode,aa_nat_list_nat(nat_list_decode,N)) = N ).

% list_decode_inverse
tff(fact_9818_list__encode__inverse,axiom,
    ! [X: list_nat] : aa_nat_list_nat(nat_list_decode,aa_list_nat_nat(nat_list_encode,X)) = X ).

% list_encode_inverse
tff(fact_9819_list__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = aa_nat_list_nat(nat_list_decode,Y) )
    <=> ( X = Y ) ) ).

% list_decode_eq
tff(fact_9820_inj__list__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on_nat_list_nat(nat_list_decode),A2)) ).

% inj_list_decode
tff(fact_9821_list__decode_Opsimps_I1_J,axiom,
    ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
   => ( aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_9822_list__decode_Osimps_I1_J,axiom,
    aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_9823_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)))
     => ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_ph,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_9824_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat(nat_list_decode,top_top_set_nat,top_top_set_list_nat) ).

% bij_list_decode
tff(fact_9825_surj__list__decode,axiom,
    image_nat_list_nat(nat_list_decode,top_top_set_nat) = top_top_set_list_nat ).

% surj_list_decode
tff(fact_9826_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_ph,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_9827_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,N2) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_ph,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
                 => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_9828_num__of__integer_Otransfer,axiom,
    bNF_re6718328864250387230um_num(code_pcr_integer,fequal_num,comp_nat_num_int(num_of_nat,nat2),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_9829_compute__powr__real,axiom,
    ! [B: real,I: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => ( aa_real_real(powr_real2(B),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_pi(B),I)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => ( ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I )
           => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(powr_real2(B),I) = aa_nat_real(power_power_real(B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))) ) )
              & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(powr_real2(B),I) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I))))) ) ) ) )
          & ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) != I )
           => ( aa_real_real(powr_real2(B),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_pi(B),I)) ) ) ) ) ) ).

% compute_powr_real
tff(fact_9830_Code__Target__Nat_ONat_Otransfer,axiom,
    bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,nat2,code_Target_Nat) ).

% Code_Target_Nat.Nat.transfer
tff(fact_9831_gcd__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),gcd_gcd_int,gcd_gcd_Code_integer) ).

% gcd_integer.transfer
tff(fact_9832_nat__of__integer_Otransfer,axiom,
    bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,nat2,code_nat_of_integer) ).

% nat_of_integer.transfer
tff(fact_9833_plus__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),plus_plus_int,plus_p5714425477246183910nteger) ).

% plus_integer.transfer
tff(fact_9834_minus__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),minus_minus_int,minus_8373710615458151222nteger) ).

% minus_integer.transfer
tff(fact_9835_less__integer_Otransfer,axiom,
    bNF_re6983685936609452663r_bool(code_pcr_integer,bNF_re3324582151780938536l_bool(code_pcr_integer,fequal_bool),ord_less_int,ord_le6747313008572928689nteger) ).

% less_integer.transfer
tff(fact_9836_uminus__integer_Otransfer,axiom,
    pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),uminus_uminus_int),uminus1351360451143612070nteger)) ).

% uminus_integer.transfer
tff(fact_9837_one__integer_Otransfer,axiom,
    pp(aa_Code_integer_bool(aa_int3865899974016033897r_bool(code_pcr_integer,one_one_int),one_one_Code_integer)) ).

% one_integer.transfer
tff(fact_9838_zero__integer_Otransfer,axiom,
    pp(aa_Code_integer_bool(aa_int3865899974016033897r_bool(code_pcr_integer,zero_zero_int),zero_z3403309356797280102nteger)) ).

% zero_integer.transfer
tff(fact_9839_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% int_of_integer_integer_of_nat
tff(fact_9840_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,X)) = aa_nat_int(semiri1314217659103216013at_int,X) ).

% integer_of_nat.rep_eq
tff(fact_9841_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_9842_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : aa_nat_Code_integer(code_integer_of_nat,X) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,X)) ).

% integer_of_nat.abs_eq
tff(fact_9843_integer__of__nat_Otransfer,axiom,
    bNF_re4153400068438556298nteger(fequal_nat,code_pcr_integer,semiri1314217659103216013at_int,code_integer_of_nat) ).

% integer_of_nat.transfer
tff(fact_9844_integer__of__nat__1,axiom,
    aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_9845_dup_Otransfer,axiom,
    pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),aTP_Lamm_ok),code_dup)) ).

% dup.transfer
tff(fact_9846_integer__of__nat__def,axiom,
    code_integer_of_nat = map_fu6290471996055670595nteger(id_nat,code_integer_of_int,semiri1314217659103216013at_int) ).

% integer_of_nat_def
tff(fact_9847_dup_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(code_dup,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,X)) ).

% dup.rep_eq
tff(fact_9848_dup_Oabs__eq,axiom,
    ! [X: int] : aa_Cod1783974652533396728nteger(code_dup,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)) ).

% dup.abs_eq
tff(fact_9849_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N))),one_one_Code_integer) ).

% Code_Numeral.sub_code(9)
tff(fact_9850_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N))),one_one_Code_integer) ).

% Code_Numeral.sub_code(8)
tff(fact_9851_sub_Orep__eq,axiom,
    ! [X: num,Xa3: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,X),Xa3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Xa3)) ).

% sub.rep_eq
tff(fact_9852_sub_Oabs__eq,axiom,
    ! [Xa3: num,X: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,Xa3),X) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Xa3)),aa_num_int(numeral_numeral_int,X))) ).

% sub.abs_eq
tff(fact_9853_sub_Otransfer,axiom,
    bNF_re4520166952630509772nteger(fequal_num,bNF_re6501075790457514782nteger(fequal_num,code_pcr_integer),aTP_Lamm_ol,code_sub) ).

% sub.transfer
tff(fact_9854_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),less_than)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_9855_elimnum,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimnum
tff(fact_9856_at__top__to__right,axiom,
    at_top_real = filtermap_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_top_to_right
tff(fact_9857_idiff__enat__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat),N) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_9858_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,extended_enat2(zero_zero_nat)) = N ).

% idiff_enat_0_right
tff(fact_9859_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] : plus_p3455044024723400733d_enat(extended_enat2(M),extended_enat2(N)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% plus_enat_simps(1)
tff(fact_9860_idiff__enat__enat,axiom,
    ! [A: nat,B: nat] : minus_3235023915231533773d_enat(extended_enat2(A),extended_enat2(B)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% idiff_enat_enat
tff(fact_9861_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ord_le72135733267957522d_enat(extended_enat2(M),extended_enat2(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% enat_ord_simps(2)
tff(fact_9862_numeral__less__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M),extended_enat2(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_less_enat_iff
tff(fact_9863_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_9864_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_9865_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_9866_one__enat__def,axiom,
    one_on7984719198319812577d_enat = extended_enat2(one_one_nat) ).

% one_enat_def
tff(fact_9867_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = one_on7984719198319812577d_enat )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(1)
tff(fact_9868_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_on7984719198319812577d_enat = extended_enat2(X) )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(2)
tff(fact_9869_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,Uu: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf(A,B),Uu) = vEBT_Leaf(A,B) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_9870_less__enatE,axiom,
    ! [N: extended_enat,M: nat] :
      ( ord_le72135733267957522d_enat(N,extended_enat2(M))
     => ~ ! [K2: nat] :
            ( ( N = extended_enat2(K2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),M)) ) ) ).

% less_enatE
tff(fact_9871_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(X,Y),extended_enat2(N))
    <=> ? [Y7: nat,X9: nat] :
          ( ( X = extended_enat2(X9) )
          & ( Y = extended_enat2(Y7) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X9),Y7)),N)) ) ) ).

% iadd_le_enat_iff
tff(fact_9872_Suc__ile__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( ord_le2932123472753598470d_enat(extended_enat2(aa_nat_nat(suc,M)),N)
    <=> ord_le72135733267957522d_enat(extended_enat2(M),N) ) ).

% Suc_ile_eq
tff(fact_9873_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(L)) = vEBT_Node(Info2,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_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))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_pj(Deg),TreeList2)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_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)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_9874_at__left__minus,axiom,
    ! [A: real] : topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)) = filtermap_real_real(uminus_uminus_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,A),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,A)))) ).

% at_left_minus
tff(fact_9875_at__right__minus,axiom,
    ! [A: real] : topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)) = filtermap_real_real(uminus_uminus_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,A),set_or5984915006950818249n_real(aa_real_real(uminus_uminus_real,A)))) ).

% at_right_minus
tff(fact_9876_at__right__to__top,axiom,
    topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)) = filtermap_real_real(inverse_inverse_real,at_top_real) ).

% at_right_to_top
tff(fact_9877_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa3) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( Y != vEBT_Leaf(A3,B3) ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
             => ( ( Xa3 = extend5688581933313929465d_enat )
               => ( Y != vEBT_Node(Info,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_pj(Deg2),TreeList),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ! [L2: nat] :
                    ( ( Xa3 = extended_enat2(L2) )
                   => ( Y != vEBT_Node(Info,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_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,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_pj(Deg2),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_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,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_9878_elimcomplete,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimcomplete
tff(fact_9879_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,extended_enat2(N)) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,extended_enat2(N)) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
tff(fact_9880_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( times_7803423173614009249d_enat(extended_enat2(M),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( times_7803423173614009249d_enat(extended_enat2(M),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
tff(fact_9881_infinity__ne__i1,axiom,
    extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).

% infinity_ne_i1
tff(fact_9882_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_9883_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A3: bool,B3: bool,Uu2: extended_enat] : X != produc581526299967858633d_enat(vEBT_Leaf(A3,B3),Uu2)
     => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X != produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extend5688581933313929465d_enat)
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,L2: nat] : X != produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extended_enat2(L2)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_9884_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info2,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_pj(Deg),TreeList2),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_9885_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa3) = Y )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(X,Xa3))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( Y = vEBT_Leaf(A3,B3) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Leaf(A3,B3),Xa3)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ( ( Xa3 = extend5688581933313929465d_enat )
                 => ( ( Y = vEBT_Node(Info,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_pj(Deg2),TreeList),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
                 => ! [L2: nat] :
                      ( ( Xa3 = extended_enat2(L2) )
                     => ( ( Y = vEBT_Node(Info,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_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,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_pj(Deg2),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_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,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extended_enat2(L2))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_9886_diff__enat__def,axiom,
    ! [A: extended_enat,B: extended_enat] : minus_3235023915231533773d_enat(A,B) = extend3600170679010898289d_enat(aTP_Lamm_pl(B),extend5688581933313929465d_enat,A) ).

% diff_enat_def
tff(fact_9887_plus__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : plus_p3455044024723400733d_enat(M,N) = extend3600170679010898289d_enat(aTP_Lamm_pn(N),extend5688581933313929465d_enat,M) ).

% plus_enat_def
tff(fact_9888_times__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : times_7803423173614009249d_enat(M,N) = extend3600170679010898289d_enat(aTP_Lamm_pp(N),if_Extended_enat(fequal_Extended_enat(N,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),M) ).

% times_enat_def
tff(fact_9889_eSuc__def,axiom,
    ! [I: extended_enat] : extended_eSuc(I) = extend3600170679010898289d_enat(aTP_Lamm_pq,extend5688581933313929465d_enat,I) ).

% eSuc_def
tff(fact_9890_Code__Numeral_Odup__def,axiom,
    code_dup = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),aTP_Lamm_ok) ).

% Code_Numeral.dup_def
tff(fact_9891_eSuc__minus__1,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(extended_eSuc(N),one_on7984719198319812577d_enat) = N ).

% eSuc_minus_1
tff(fact_9892_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( extended_enat2(Y) = extended_eSuc(X) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( extended_enat2(N4) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_9893_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( extended_eSuc(X) = extended_enat2(Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( X = extended_enat2(N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_9894_eSuc__enat,axiom,
    ! [N: nat] : extended_eSuc(extended_enat2(N)) = extended_enat2(aa_nat_nat(suc,N)) ).

% eSuc_enat
tff(fact_9895_one__eSuc,axiom,
    one_on7984719198319812577d_enat = extended_eSuc(zero_z5237406670263579293d_enat) ).

% one_eSuc
tff(fact_9896_eSuc__plus__1,axiom,
    ! [N: extended_enat] : extended_eSuc(N) = plus_p3455044024723400733d_enat(N,one_on7984719198319812577d_enat) ).

% eSuc_plus_1
tff(fact_9897_plus__1__eSuc_I1_J,axiom,
    ! [Q4: extended_enat] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,Q4) = extended_eSuc(Q4) ).

% plus_1_eSuc(1)
tff(fact_9898_plus__1__eSuc_I2_J,axiom,
    ! [Q4: extended_enat] : plus_p3455044024723400733d_enat(Q4,one_on7984719198319812577d_enat) = extended_eSuc(Q4) ).

% plus_1_eSuc(2)
tff(fact_9899_uminus__integer__def,axiom,
    uminus1351360451143612070nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),uminus_uminus_int) ).

% uminus_integer_def
tff(fact_9900_gcd__integer__def,axiom,
    gcd_gcd_Code_integer = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),gcd_gcd_int) ).

% gcd_integer_def
tff(fact_9901_minus__integer__def,axiom,
    minus_8373710615458151222nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),minus_minus_int) ).

% minus_integer_def
tff(fact_9902_plus__integer__def,axiom,
    plus_p5714425477246183910nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),plus_plus_int) ).

% plus_integer_def
tff(fact_9903_has__vector__derivative__id,axiom,
    ! [Net: filter_real] : has_ve631408500373753343e_real(aTP_Lamm_ba,one_one_real,Net) ).

% has_vector_derivative_id
tff(fact_9904_Quotient__real,axiom,
    quotie6870213296501934825t_real(realrel,real2,rep_real,cr_real) ).

% Quotient_real
tff(fact_9905_inj__on__char__of__nat,axiom,
    pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_9906_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9907_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9908_char_Osize_I2_J,axiom,
    ! [X1: bool,X2: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : size_size_char(char2(X1,X2,X32,X42,X52,X62,X72,X82)) = zero_zero_nat ).

% char.size(2)
tff(fact_9909_nat__of__char__less__256,axiom,
    ! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% nat_of_char_less_256
tff(fact_9910_char_Osize__gen,axiom,
    ! [X1: bool,X2: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : size_char(char2(X1,X2,X32,X42,X52,X62,X72,X82)) = zero_zero_nat ).

% char.size_gen
tff(fact_9911_cr__int__def,axiom,
    ! [X4: product_prod_nat_nat] : aa_Pro2928840338864870335t_bool(cr_int,X4) = aa_int_fun_int_bool(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X4)) ).

% cr_int_def
tff(fact_9912_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_9913_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi6264000038957366511cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% less_eq
tff(fact_9914_Quotient__int,axiom,
    quotie1194848508323700631at_int(intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_9915_gcd__nat_Osemilattice__neutr__axioms,axiom,
    pp(aa_nat_bool(semila9081495762789891438tr_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_9916_max__nat_Osemilattice__neutr__axioms,axiom,
    pp(aa_nat_bool(semila9081495762789891438tr_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_9917_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_9918_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N)),aTP_Lamm_pr(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N))) ).

% Restr_natLeq
tff(fact_9919_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_pt(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N))) ).

% Restr_natLeq2
tff(fact_9920_rcis__cnj,axiom,
    ! [A: complex] : aa_complex_complex(cnj,A) = aa_real_complex(rcis(aa_complex_real(real_V1022390504157884413omplex,A)),aa_real_real(uminus_uminus_real,aa_complex_real(arg,A))) ).

% rcis_cnj
tff(fact_9921_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_9922_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_9923_complex__cnj__minus,axiom,
    ! [X: complex] : aa_complex_complex(cnj,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(cnj,X)) ).

% complex_cnj_minus
tff(fact_9924_complex__cnj__inverse,axiom,
    ! [X: complex] : aa_complex_complex(cnj,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cnj,X)) ).

% complex_cnj_inverse
tff(fact_9925_complex__cnj__of__int,axiom,
    ! [Z: int] : aa_complex_complex(cnj,aa_int_complex(ring_17405671764205052669omplex,Z)) = aa_int_complex(ring_17405671764205052669omplex,Z) ).

% complex_cnj_of_int
tff(fact_9926_complex__cnj__i,axiom,
    aa_complex_complex(cnj,imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% complex_cnj_i
tff(fact_9927_complex__cnj__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_cnj_neg_numeral
tff(fact_9928_natLeq__underS__less,axiom,
    ! [N: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N)) ).

% natLeq_underS_less
tff(fact_9929_cnj_Osimps_I2_J,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(cnj,Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% cnj.simps(2)
tff(fact_9930_complex__cnj,axiom,
    ! [A: real,B: real] : aa_complex_complex(cnj,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(A),aa_real_real(uminus_uminus_real,B)) ).

% complex_cnj
tff(fact_9931_cis__cnj,axiom,
    ! [T: real] : aa_complex_complex(cnj,aa_real_complex(cis,T)) = aa_real_complex(cis,aa_real_real(uminus_uminus_real,T)) ).

% cis_cnj
tff(fact_9932_cnj_Ocode,axiom,
    ! [Z: complex] : aa_complex_complex(cnj,Z) = aa_real_complex(complex2(aa_complex_real(re,Z)),aa_real_real(uminus_uminus_real,aa_complex_real(im,Z))) ).

% cnj.code
tff(fact_9933_gcd__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.monoid_axioms
tff(fact_9934_max__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.monoid_axioms
tff(fact_9935_Lcm__eq__0__I__nat,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( aa_set_nat_nat(gcd_Lcm_nat,A2) = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
tff(fact_9936_abs__Lcm__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Lcm_int(K5)) = gcd_Lcm_int(K5) ).

% abs_Lcm_eq
tff(fact_9937_Lcm__0__iff__nat,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(gcd_Lcm_nat,A2) = zero_zero_nat )
      <=> pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% Lcm_0_iff_nat
tff(fact_9938_Lcm__abs__eq,axiom,
    ! [K5: set_int] : gcd_Lcm_int(image_int_int(abs_abs_int,K5)) = gcd_Lcm_int(K5) ).

% Lcm_abs_eq
tff(fact_9939_Lcm__int__eq,axiom,
    ! [N5: set_nat] : gcd_Lcm_int(image_nat_int(semiri1314217659103216013at_int,N5)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,N5)) ).

% Lcm_int_eq
tff(fact_9940_Lcm__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(aTP_Lamm_lm,K5)) = aa_int_nat(nat2,gcd_Lcm_int(K5)) ).

% Lcm_nat_abs_eq
tff(fact_9941_Lcm__nat__infinite,axiom,
    ! [M10: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,M10))
     => ( aa_set_nat_nat(gcd_Lcm_nat,M10) = zero_zero_nat ) ) ).

% Lcm_nat_infinite
tff(fact_9942_Lcm__least__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B3: int] :
          ( member_int(B3,A2)
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),A)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,gcd_Lcm_int(A2)),A)) ) ).

% Lcm_least_int
tff(fact_9943_dvd__Lcm__int,axiom,
    ! [M: int,M10: set_int] :
      ( member_int(M,M10)
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),gcd_Lcm_int(M10))) ) ).

% dvd_Lcm_int
tff(fact_9944_dvd__Lcm__nat,axiom,
    ! [M: nat,M10: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(M),M10))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_set_nat_nat(gcd_Lcm_nat,M10))) ) ).

% dvd_Lcm_nat
tff(fact_9945_Lcm__dvd__nat,axiom,
    ! [M10: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),M10))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X3),N)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(gcd_Lcm_nat,M10)),N)) ) ).

% Lcm_dvd_nat
tff(fact_9946_Lcm__int__greater__eq__0,axiom,
    ! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Lcm_int(K5))) ).

% Lcm_int_greater_eq_0
tff(fact_9947_Lcm__nat__empty,axiom,
    aa_set_nat_nat(gcd_Lcm_nat,bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_9948_Gcd__nat__def,axiom,
    ! [M10: set_nat] : aa_set_nat_nat(gcd_Gcd_nat,M10) = aa_set_nat_nat(gcd_Lcm_nat,collect_nat(aTP_Lamm_pu(M10))) ).

% Gcd_nat_def
tff(fact_9949_Lcm__int__def,axiom,
    ! [K5: set_int] : gcd_Lcm_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Lcm_int_def
tff(fact_9950_Lcm__eq__Max__nat,axiom,
    ! [M10: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M10))
     => ( ( M10 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M10))
         => ( ! [M2: nat,N2: nat] :
                ( pp(aa_set_nat_bool(member_nat(M2),M10))
               => ( pp(aa_set_nat_bool(member_nat(N2),M10))
                 => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N2)),M10)) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M10) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M10) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_9951_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Lcm_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_lcm_int,Xs),one_one_int) ).

% Lcm_int_set_eq_fold
tff(fact_9952_lcm__0__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = zero_zero_int )
    <=> ( ( M = zero_zero_int )
        | ( N = zero_zero_int ) ) ) ).

% lcm_0_iff_int
tff(fact_9953_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
tff(fact_9954_abs__lcm__int,axiom,
    ! [I: int,J: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,I),J)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,I),J) ).

% abs_lcm_int
tff(fact_9955_lcm__abs1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(abs_abs_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) ).

% lcm_abs1_int
tff(fact_9956_lcm__abs2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),aa_int_int(abs_abs_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) ).

% lcm_abs2_int
tff(fact_9957_lcm__proj2__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,X),Y) = Y ) ) ).

% lcm_proj2_if_dvd_nat
tff(fact_9958_lcm__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Y),X) = Y ) ) ).

% lcm_proj1_if_dvd_nat
tff(fact_9959_lcm__proj2__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = N )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% lcm_proj2_iff_nat
tff(fact_9960_lcm__proj1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% lcm_proj1_iff_nat
tff(fact_9961_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% lcm_1_iff_nat
tff(fact_9962_lcm__1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) )
        & ( ( N = one_one_int )
          | ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_9963_lcm__proj1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = aa_int_int(abs_abs_int,M) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ).

% lcm_proj1_iff_int
tff(fact_9964_lcm__proj2__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N) = aa_int_int(abs_abs_int,N) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ).

% lcm_proj2_iff_int
tff(fact_9965_lcm__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Y),X) = aa_int_int(abs_abs_int,Y) ) ) ).

% lcm_proj1_if_dvd_int
tff(fact_9966_lcm__proj2__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) = aa_int_int(abs_abs_int,Y) ) ) ).

% lcm_proj2_if_dvd_int
tff(fact_9967_lcm__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ).

% lcm_int_int_eq
tff(fact_9968_lcm__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ).

% lcm_nat_abs_right_eq
tff(fact_9969_lcm__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% lcm_nat_abs_left_eq
tff(fact_9970_Lcm__nat__insert,axiom,
    ! [N: nat,M10: set_nat] : aa_set_nat_nat(gcd_Lcm_nat,insert_nat(N,M10)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,N),aa_set_nat_nat(gcd_Lcm_nat,M10)) ).

% Lcm_nat_insert
tff(fact_9971_lcm__nat__def,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).

% lcm_nat_def
tff(fact_9972_lcm__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),D))
        & ! [E3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),E3))
              & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),E3)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),E3)) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,A),B) ) ) ).

% lcm_unique_int
tff(fact_9973_prod__gcd__lcm__nat,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ).

% prod_gcd_lcm_nat
tff(fact_9974_lcm__neg1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) ).

% lcm_neg1_int
tff(fact_9975_lcm__neg2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) ).

% lcm_neg2_int
tff(fact_9976_lcm__int__def,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).

% lcm_int_def
tff(fact_9977_lcm__unique__nat,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),D))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),D))
        & ! [E3: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),E3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),E3)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),E3)) ) )
    <=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,A),B) ) ) ).

% lcm_unique_nat
tff(fact_9978_dvd__lcm__I2__nat,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N))) ) ).

% dvd_lcm_I2_nat
tff(fact_9979_dvd__lcm__I1__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N))) ) ).

% dvd_lcm_I1_nat
tff(fact_9980_dvd__lcm__I2__int,axiom,
    ! [I: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,I),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,I),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N))) ) ).

% dvd_lcm_I2_int
tff(fact_9981_dvd__lcm__I1__int,axiom,
    ! [I: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,I),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,I),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N))) ) ).

% dvd_lcm_I1_int
tff(fact_9982_lcm__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),gcd_lcm_int,gcd_lcm_int) ).

% lcm_integer.rsp
tff(fact_9983_lcm__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_bool] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y))) ) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y))) ) ) ) ) ).

% lcm_cases_int
tff(fact_9984_lcm__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,X),Y))) ).

% lcm_ge_0_int
tff(fact_9985_lcm__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( N != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N))) ) ) ).

% lcm_pos_int
tff(fact_9986_lcm__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N))) ) ) ).

% lcm_pos_nat
tff(fact_9987_prod__gcd__lcm__int,axiom,
    ! [M: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,M)),aa_int_int(abs_abs_int,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M),N)) ).

% prod_gcd_lcm_int
tff(fact_9988_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M10: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M10))
     => ( ( M10 != bot_bot_set_nat )
       => ( ! [M2: nat,N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(M2),M10))
             => ( pp(aa_set_nat_bool(member_nat(N2),M10))
               => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N2)),M10)) ) )
         => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Lcm_nat,M10)),M10)) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_9989_lcm__altdef__int,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,A),B) = aa_int_int(divide_divide_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(gcd_gcd_int,A),B)) ).

% lcm_altdef_int
tff(fact_9990_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Lcm_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_lcm_nat,Xs),one_one_nat) ).

% Lcm_nat_set_eq_fold
tff(fact_9991_Lcm__nat__def,axiom,
    ! [M10: set_nat] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,M10))
       => ( aa_set_nat_nat(gcd_Lcm_nat,M10) = aa_set_nat_nat(lattic7826324295020591184_F_nat(gcd_lcm_nat,one_one_nat),M10) ) )
      & ( ~ pp(aa_set_nat_bool(finite_finite_nat,M10))
       => ( aa_set_nat_nat(gcd_Lcm_nat,M10) = zero_zero_nat ) ) ) ).

% Lcm_nat_def
tff(fact_9992_lcm__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,X),Xa3)) = aa_int_int(aa_int_fun_int_int(gcd_lcm_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa3)) ).

% lcm_integer.rep_eq
tff(fact_9993_lcm__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa3)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(gcd_lcm_int,Xa3),X)) ).

% lcm_integer.abs_eq
tff(fact_9994_lcm__code__integer,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,A),B) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,A),B)) ).

% lcm_code_integer
tff(fact_9995_lcm__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),gcd_lcm_int,gcd_lcm_Code_integer) ).

% lcm_integer.transfer
tff(fact_9996_lcm__integer__def,axiom,
    gcd_lcm_Code_integer = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),gcd_lcm_int) ).

% lcm_integer_def
tff(fact_9997_unit__factor__simps_I1_J,axiom,
    aa_nat_nat(unit_f2748546683901255202or_nat,zero_zero_nat) = zero_zero_nat ).

% unit_factor_simps(1)
tff(fact_9998_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : aa_nat_nat(unit_f2748546683901255202or_nat,aa_nat_nat(suc,N)) = one_one_nat ).

% unit_factor_simps(2)
tff(fact_9999_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = one_one_nat ) ) ) ).

% unit_factor_nat_def
tff(fact_10000_gcd__nat_Ocomm__monoid__axioms,axiom,
    pp(aa_nat_bool(comm_monoid_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.comm_monoid_axioms
tff(fact_10001_max__nat_Ocomm__monoid__axioms,axiom,
    pp(aa_nat_bool(comm_monoid_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.comm_monoid_axioms
tff(fact_10002_times__num__def,axiom,
    ! [M: num,N: num] : aa_num_num(times_times_num(M),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ).

% times_num_def
tff(fact_10003_nat__of__num__mult,axiom,
    ! [X: num,Y: num] : aa_num_nat(nat_of_num,aa_num_num(times_times_num(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,Y)) ).

% nat_of_num_mult
tff(fact_10004_nat__of__num__sqr,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(sqr,X)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X)) ).

% nat_of_num_sqr
tff(fact_10005_nat__of__num__inc,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(inc,X)) = aa_nat_nat(suc,aa_num_nat(nat_of_num,X)) ).

% nat_of_num_inc
tff(fact_10006_nat__of__num__inverse,axiom,
    ! [X: num] : aa_nat_num(num_of_nat,aa_num_nat(nat_of_num,X)) = X ).

% nat_of_num_inverse
tff(fact_10007_num__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa_num_nat(nat_of_num,X) = aa_num_nat(nat_of_num,Y) ) ) ).

% num_eq_iff
tff(fact_10008_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral_nat ).

% nat_of_num_numeral
tff(fact_10009_less__num__def,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ) ).

% less_num_def
tff(fact_10010_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(nat_of_num,X))) ).

% nat_of_num_pos
tff(fact_10011_nat__of__num__neq__0,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,X) != zero_zero_nat ).

% nat_of_num_neq_0
tff(fact_10012_nat__of__num__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,N)),aa_num_nat(nat_of_num,N)) ).

% nat_of_num_code(2)
tff(fact_10013_nat__of__num__code_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = one_one_nat ).

% nat_of_num_code(1)
tff(fact_10014_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X)) ).

% nat_of_num.simps(2)
tff(fact_10015_nat__of__num__add,axiom,
    ! [X: num,Y: num] : aa_num_nat(nat_of_num,aa_num_num(plus_plus_num(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,Y)) ).

% nat_of_num_add
tff(fact_10016_less__eq__num__def,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ) ).

% less_eq_num_def
tff(fact_10017_nat__of__num_Osimps_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_of_num.simps(1)
tff(fact_10018_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,X)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X))) ).

% nat_of_num.simps(3)
tff(fact_10019_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(nat_of_num,aa_nat_num(num_of_nat,N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_10020_nat__of__num__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,N)),aa_num_nat(nat_of_num,N))) ).

% nat_of_num_code(3)
tff(fact_10021_plus__num__def,axiom,
    ! [M: num,N: num] : aa_num_num(plus_plus_num(M),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ).

% plus_num_def
tff(fact_10022_real__floor__code,axiom,
    ! [X: rat] : aa_real_int(archim6058952711729229775r_real,aa_rat_real(ratreal,X)) = aa_rat_int(archim3151403230148437115or_rat,X) ).

% real_floor_code
tff(fact_10023_real__minus__code,axiom,
    ! [X: rat,Y: rat] : aa_real_real(minus_minus_real(aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)) = aa_rat_real(ratreal,aa_rat_rat(minus_minus_rat(X),Y)) ).

% real_minus_code
tff(fact_10024_real__less__eq__code,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% real_less_eq_code
tff(fact_10025_real__less__code,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% real_less_code
tff(fact_10026_zero__real__code,axiom,
    zero_zero_real = aa_rat_real(ratreal,zero_zero_rat) ).

% zero_real_code
tff(fact_10027_Ratreal__def,axiom,
    ratreal = field_7254667332652039916t_real ).

% Ratreal_def
tff(fact_10028_real__uminus__code,axiom,
    ! [X: rat] : aa_real_real(uminus_uminus_real,aa_rat_real(ratreal,X)) = aa_rat_real(ratreal,aa_rat_rat(uminus_uminus_rat,X)) ).

% real_uminus_code
tff(fact_10029_one__real__code,axiom,
    one_one_real = aa_rat_real(ratreal,one_one_rat) ).

% one_real_code
tff(fact_10030_real__times__code,axiom,
    ! [X: rat,Y: rat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)) = aa_rat_real(ratreal,aa_rat_rat(times_times_rat(X),Y)) ).

% real_times_code
tff(fact_10031_real__plus__code,axiom,
    ! [X: rat,Y: rat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)) = aa_rat_real(ratreal,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) ).

% real_plus_code
tff(fact_10032_real__inverse__code,axiom,
    ! [X: rat] : aa_real_real(inverse_inverse_real,aa_rat_real(ratreal,X)) = aa_rat_real(ratreal,aa_rat_rat(inverse_inverse_rat,X)) ).

% real_inverse_code
tff(fact_10033_real__divide__code,axiom,
    ! [X: rat,Y: rat] : aa_real_real(divide_divide_real(aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)) = aa_rat_real(ratreal,aa_rat_rat(divide_divide_rat(X),Y)) ).

% real_divide_code
tff(fact_10034_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_We3818239936649020644el_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N)))) ).

% natLeq_on_wo_rel
tff(fact_10035_gcd__nat_Osemilattice__order__axioms,axiom,
    semila1248733672344298208er_nat(gcd_gcd_nat,dvd_dvd_nat,aTP_Lamm_ou) ).

% gcd_nat.semilattice_order_axioms
tff(fact_10036_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,N)),aa_nat_nat(suc,zero_zero_nat))),aa_Code_natural_nat(code_nat_of_natural,N))) ) ).

% natural_decr
tff(fact_10037_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Xa3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa3)) ).

% plus_natural.rep_eq
tff(fact_10038_minus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,X),Xa3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa3)) ).

% minus_natural.rep_eq
tff(fact_10039_one__natural_Orep__eq,axiom,
    aa_Code_natural_nat(code_nat_of_natural,one_one_Code_natural) = one_one_nat ).

% one_natural.rep_eq
tff(fact_10040_zero__natural_Orep__eq,axiom,
    aa_Code_natural_nat(code_nat_of_natural,zero_z2226904508553997617atural) = zero_zero_nat ).

% zero_natural.rep_eq
tff(fact_10041_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa3: code_natural] :
      ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,X),Xa3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa3))) ) ).

% less_natural.rep_eq
tff(fact_10042_natural__zero__minus__one,axiom,
    aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,zero_z2226904508553997617atural),one_one_Code_natural) = zero_z2226904508553997617atural ).

% natural_zero_minus_one
tff(fact_10043_integer__of__natural_Orep__eq,axiom,
    ! [X: code_natural] : aa_Code_integer_int(code_int_of_integer,aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,X)) = aa_nat_int(semiri1314217659103216013at_int,aa_Code_natural_nat(code_nat_of_natural,X)) ).

% integer_of_natural.rep_eq
tff(fact_10044_int__of__integer__of__natural,axiom,
    ! [N: code_natural] : aa_Code_integer_int(code_int_of_integer,aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_Code_natural_nat(code_nat_of_natural,N)) ).

% int_of_integer_of_natural
tff(fact_10045_log_Oelims,axiom,
    ! [X: code_natural,Xa3: code_natural,Y: code_natural] :
      ( ( log(X,Xa3) = Y )
     => ( ( ( ord_le1926595141338095240atural(X,one_one_Code_natural)
            | pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa3),X)) )
         => ( Y = one_one_Code_natural ) )
        & ( ~ ( ord_le1926595141338095240atural(X,one_one_Code_natural)
              | pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa3),X)) )
         => ( Y = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,divide5121882707175180666atural(Xa3,X))) ) ) ) ) ).

% log.elims
tff(fact_10046_log_Osimps,axiom,
    ! [B: code_natural,I: code_natural] :
      ( ( ( ord_le1926595141338095240atural(B,one_one_Code_natural)
          | pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,I),B)) )
       => ( log(B,I) = one_one_Code_natural ) )
      & ( ~ ( ord_le1926595141338095240atural(B,one_one_Code_natural)
            | pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,I),B)) )
       => ( log(B,I) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(B,divide5121882707175180666atural(I,B))) ) ) ) ).

% log.simps
tff(fact_10047_next_Osimps,axiom,
    ! [V: code_natural,W: code_natural] : aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W)) = aa_Pro7165575656943818355atural(produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,minus_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(V,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(V,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))),one_one_Code_natural))),one_one_Code_natural)),produc3574140220909816553atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(V,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(V,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))) ).

% next.simps
tff(fact_10048_Code__Numeral_ONat_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod607569804290114243atural(code_Nat,X)) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% Code_Numeral.Nat.rep_eq
tff(fact_10049_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),K),aTP_Lamm_pw,one_one_Code_natural),aTP_Lamm_px(K)) ).

% Random.range_def
tff(fact_10050_inc__shift__def,axiom,
    ! [V: code_natural,K: code_natural] :
      ( ( ( V = K )
       => ( inc_shift(V,K) = one_one_Code_natural ) )
      & ( ( V != K )
       => ( inc_shift(V,K) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,K),one_one_Code_natural) ) ) ) ).

% inc_shift_def
tff(fact_10051_log_Opelims,axiom,
    ! [X: code_natural,Xa3: code_natural,Y: code_natural] :
      ( ( log(X,Xa3) = Y )
     => ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa3))
       => ~ ( ( ( ( ord_le1926595141338095240atural(X,one_one_Code_natural)
                  | pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa3),X)) )
               => ( Y = one_one_Code_natural ) )
              & ( ~ ( ord_le1926595141338095240atural(X,one_one_Code_natural)
                    | pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa3),X)) )
               => ( Y = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,divide5121882707175180666atural(Xa3,X))) ) ) )
           => ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa3)) ) ) ) ).

% log.pelims
tff(fact_10052_natural__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod607569804290114243atural(code_n4118661773612635043nteger,X)) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% natural_of_integer.rep_eq
tff(fact_10053_integer__of__natural__def,axiom,
    code_i5400310926305786745atural = map_fu2787874002554666395nteger(code_nat_of_natural,code_integer_of_int,semiri1314217659103216013at_int) ).

% integer_of_natural_def
tff(fact_10054_zero__natural__def,axiom,
    zero_z2226904508553997617atural = aa_nat_Code_natural(code_natural_of_nat,zero_zero_nat) ).

% zero_natural_def
tff(fact_10055_less__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] :
      ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(code_natural_of_nat,Xa3)),aa_nat_Code_natural(code_natural_of_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa3),X)) ) ).

% less_natural.abs_eq
tff(fact_10056_plus__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(code_natural_of_nat,Xa3)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa3),X)) ).

% plus_natural.abs_eq
tff(fact_10057_minus__natural_Oabs__eq,axiom,
    ! [Xa3: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,aa_nat_Code_natural(code_natural_of_nat,Xa3)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa3),X)) ).

% minus_natural.abs_eq
tff(fact_10058_one__natural__def,axiom,
    one_one_Code_natural = aa_nat_Code_natural(code_natural_of_nat,one_one_nat) ).

% one_natural_def
tff(fact_10059_integer__of__natural_Oabs__eq,axiom,
    ! [X: nat] : aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,X)) ).

% integer_of_natural.abs_eq
tff(fact_10060_natural__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Cod607569804290114243atural(code_n4118661773612635043nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_int_nat(nat2,X)) ).

% natural_of_integer.abs_eq
tff(fact_10061_Code__Numeral_ONat_Oabs__eq,axiom,
    ! [X: int] : aa_Cod607569804290114243atural(code_Nat,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_int_nat(nat2,X)) ).

% Code_Numeral.Nat.abs_eq
tff(fact_10062_Code__Numeral_ONat__def,axiom,
    code_Nat = map_fu1051355602067684763atural(code_int_of_integer,code_natural_of_nat,nat2) ).

% Code_Numeral.Nat_def
tff(fact_10063_natural__of__integer__def,axiom,
    code_n4118661773612635043nteger = map_fu1051355602067684763atural(code_int_of_integer,code_natural_of_nat,nat2) ).

% natural_of_integer_def
tff(fact_10064_Suc_Orep__eq,axiom,
    ! [X: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(code_Suc,X)) = aa_nat_nat(suc,aa_Code_natural_nat(code_nat_of_natural,X)) ).

% Suc.rep_eq
tff(fact_10065_Code__Numeral_ONat_Otransfer,axiom,
    bNF_re1831474436612530402atural(code_pcr_integer,code_pcr_natural,nat2,code_Nat) ).

% Code_Numeral.Nat.transfer
tff(fact_10066_zero__natural_Otransfer,axiom,
    pp(aa_Code_natural_bool(aa_nat4473165201473010874l_bool(code_pcr_natural,zero_zero_nat),zero_z2226904508553997617atural)) ).

% zero_natural.transfer
tff(fact_10067_Suc_Otransfer,axiom,
    pp(aa_fun857525882259921228l_bool(aa_fun1838662137086867784l_bool(bNF_re3704215830270325841atural(code_pcr_natural,code_pcr_natural),suc),code_Suc)) ).

% Suc.transfer
tff(fact_10068_less__natural_Otransfer,axiom,
    bNF_re5766837079853173503l_bool(code_pcr_natural,bNF_re8879404696352373527l_bool(code_pcr_natural,fequal_bool),ord_less_nat,ord_le5570908160329646204atural) ).

% less_natural.transfer
tff(fact_10069_minus__natural_Otransfer,axiom,
    bNF_re1400205106991740995atural(code_pcr_natural,bNF_re3704215830270325841atural(code_pcr_natural,code_pcr_natural),minus_minus_nat,minus_7197305767214868737atural) ).

% minus_natural.transfer
tff(fact_10070_plus__natural_Otransfer,axiom,
    bNF_re1400205106991740995atural(code_pcr_natural,bNF_re3704215830270325841atural(code_pcr_natural,code_pcr_natural),plus_plus_nat,plus_p4538020629002901425atural) ).

% plus_natural.transfer
tff(fact_10071_one__natural_Otransfer,axiom,
    pp(aa_Code_natural_bool(aa_nat4473165201473010874l_bool(code_pcr_natural,one_one_nat),one_one_Code_natural)) ).

% one_natural.transfer
tff(fact_10072_Suc_Oabs__eq,axiom,
    ! [X: nat] : aa_Cod4272041906116450446atural(code_Suc,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(suc,X)) ).

% Suc.abs_eq
tff(fact_10073_integer__of__natural_Otransfer,axiom,
    bNF_re5252274238750452962nteger(code_pcr_natural,code_pcr_integer,semiri1314217659103216013at_int,code_i5400310926305786745atural) ).

% integer_of_natural.transfer
tff(fact_10074_natural__of__integer_Otransfer,axiom,
    bNF_re1831474436612530402atural(code_pcr_integer,code_pcr_natural,nat2,code_n4118661773612635043nteger) ).

% natural_of_integer.transfer
tff(fact_10075_Suc__natural__minus__one,axiom,
    ! [N: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,aa_Cod4272041906116450446atural(code_Suc,N)),one_one_Code_natural) = N ).

% Suc_natural_minus_one
tff(fact_10076_Code__Numeral_OSuc__def,axiom,
    code_Suc = aa_fun5114738747585492520atural(map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),suc) ).

% Code_Numeral.Suc_def
tff(fact_10077_Code__Numeral_Osub__def,axiom,
    code_sub = map_fu8540202848259802501nteger(id_num,map_fu8638147718074629079nteger(id_num,code_integer_of_int),aTP_Lamm_ol) ).

% Code_Numeral.sub_def
tff(fact_10078_minus__natural__def,axiom,
    minus_7197305767214868737atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),minus_minus_nat) ).

% minus_natural_def
tff(fact_10079_plus__natural__def,axiom,
    plus_p4538020629002901425atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),plus_plus_nat) ).

% plus_natural_def
tff(fact_10080_Quotient3__int,axiom,
    quotie6776551016481293500at_int(intrel,abs_Integ,rep_Integ) ).

% Quotient3_int
tff(fact_10081_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N)) ).

% Field_natLeq_on
tff(fact_10082_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N)))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N)))) ).

% natLeq_on_Well_order
tff(fact_10083_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_10084_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_10085_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_10086_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,N)),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ps(N)))) ).

% natLeq_on_well_order_on
tff(fact_10087_less__eq__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(neg,K)),aa_num_int(neg,L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),K)) ) ).

% less_eq_int_code(9)
tff(fact_10088_less__eq__int__code_I7_J,axiom,
    ! [K: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(neg,K)),zero_zero_int)) ).

% less_eq_int_code(7)
tff(fact_10089_less__eq__int__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(neg,L))) ).

% less_eq_int_code(3)
tff(fact_10090_nat__code_I1_J,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(neg,K)) = zero_zero_nat ).

% nat_code(1)
tff(fact_10091_less__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(neg,K)),aa_num_int(neg,L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,L),K)) ) ).

% less_int_code(9)
tff(fact_10092_plus__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(neg,M)),aa_num_int(neg,N)) = aa_num_int(neg,aa_num_num(plus_plus_num(M),N)) ).

% plus_int_code(6)
tff(fact_10093_less__int__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(neg,L))) ).

% less_int_code(3)
tff(fact_10094_less__int__code_I7_J,axiom,
    ! [K: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(neg,K)),zero_zero_int)) ).

% less_int_code(7)
tff(fact_10095_Int_Osub__code_I4_J,axiom,
    ! [N: num] : aa_num_int(sub(one),aa_num_num(bit0,N)) = aa_num_int(neg,aa_num_num(bitM,N)) ).

% Int.sub_code(4)
tff(fact_10096_Int_Osub__code_I5_J,axiom,
    ! [N: num] : aa_num_int(sub(one),aa_num_num(bit1,N)) = aa_num_int(neg,aa_num_num(bit0,N)) ).

% Int.sub_code(5)
tff(fact_10097_Int_Osub__code_I1_J,axiom,
    aa_num_int(sub(one),one) = zero_zero_int ).

% Int.sub_code(1)
tff(fact_10098_Int_Osub__def,axiom,
    ! [M: num,N: num] : aa_num_int(sub(M),N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% Int.sub_def
tff(fact_10099_minus__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(neg,M)),aa_num_int(neg,N)) = aa_num_int(sub(N),M) ).

% minus_int_code(6)
tff(fact_10100_Int_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(dup,aa_num_int(sub(M),N))),one_one_int) ).

% Int.sub_code(9)
tff(fact_10101_Int_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(dup,aa_num_int(sub(M),N))),one_one_int) ).

% Int.sub_code(8)
tff(fact_10102_Int_Odup__code_I1_J,axiom,
    aa_int_int(dup,zero_zero_int) = zero_zero_int ).

% Int.dup_code(1)
tff(fact_10103_Int_Odup__def,axiom,
    ! [K: int] : aa_int_int(dup,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),K) ).

% Int.dup_def
tff(fact_10104_Int_Odup__code_I3_J,axiom,
    ! [N: num] : aa_int_int(dup,aa_num_int(neg,N)) = aa_num_int(neg,aa_num_num(bit0,N)) ).

% Int.dup_code(3)
tff(fact_10105_Int_Osub__code_I6_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_int_int(dup,aa_num_int(sub(M),N)) ).

% Int.sub_code(6)
tff(fact_10106_Int_Osub__code_I7_J,axiom,
    ! [M: num,N: num] : aa_num_int(sub(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_int_int(dup,aa_num_int(sub(M),N)) ).

% Int.sub_code(7)
tff(fact_10107_Int_Osub__code_I2_J,axiom,
    ! [M: num] : aa_num_int(sub(aa_num_num(bit0,M)),one) = aa_num_int(pos,aa_num_num(bitM,M)) ).

% Int.sub_code(2)
tff(fact_10108_Int_Osub__code_I3_J,axiom,
    ! [M: num] : aa_num_int(sub(aa_num_num(bit1,M)),one) = aa_num_int(pos,aa_num_num(bit0,M)) ).

% Int.sub_code(3)
tff(fact_10109_Int_Odup__code_I2_J,axiom,
    ! [N: num] : aa_int_int(dup,aa_num_int(pos,N)) = aa_num_int(pos,aa_num_num(bit0,N)) ).

% Int.dup_code(2)
tff(fact_10110_less__eq__int__code_I8_J,axiom,
    ! [K: num,L: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(neg,K)),aa_num_int(pos,L))) ).

% less_eq_int_code(8)
tff(fact_10111_less__eq__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(pos,K)),aa_num_int(neg,L))) ).

% less_eq_int_code(6)
tff(fact_10112_uminus__int__code_I3_J,axiom,
    ! [M: num] : aa_int_int(uminus_uminus_int,aa_num_int(neg,M)) = aa_num_int(pos,M) ).

% uminus_int_code(3)
tff(fact_10113_uminus__int__code_I2_J,axiom,
    ! [M: num] : aa_int_int(uminus_uminus_int,aa_num_int(pos,M)) = aa_num_int(neg,M) ).

% uminus_int_code(2)
tff(fact_10114_Int_ONeg__def,axiom,
    ! [N: num] : aa_num_int(neg,N) = aa_int_int(uminus_uminus_int,aa_num_int(pos,N)) ).

% Int.Neg_def
tff(fact_10115_less__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(pos,K)),aa_num_int(neg,L))) ).

% less_int_code(6)
tff(fact_10116_less__int__code_I8_J,axiom,
    ! [K: num,L: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(neg,K)),aa_num_int(pos,L))) ).

% less_int_code(8)
tff(fact_10117_less__int__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(pos,K)),zero_zero_int)) ).

% less_int_code(4)
tff(fact_10118_less__int__code_I2_J,axiom,
    ! [L: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(pos,L))) ).

% less_int_code(2)
tff(fact_10119_one__int__code,axiom,
    one_one_int = aa_num_int(pos,one) ).

% one_int_code
tff(fact_10120_plus__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(pos,M)),aa_num_int(pos,N)) = aa_num_int(pos,aa_num_num(plus_plus_num(M),N)) ).

% plus_int_code(3)
tff(fact_10121_less__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(pos,K)),aa_num_int(pos,L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,K),L)) ) ).

% less_int_code(5)
tff(fact_10122_nat__code_I3_J,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(pos,K)) = aa_num_nat(nat_of_num,K) ).

% nat_code(3)
tff(fact_10123_less__eq__int__code_I2_J,axiom,
    ! [L: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(pos,L))) ).

% less_eq_int_code(2)
tff(fact_10124_less__eq__int__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(pos,K)),zero_zero_int)) ).

% less_eq_int_code(4)
tff(fact_10125_less__eq__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(pos,K)),aa_num_int(pos,L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,K),L)) ) ).

% less_eq_int_code(5)
tff(fact_10126_Int_OPos__def,axiom,
    pos = numeral_numeral_int ).

% Int.Pos_def
tff(fact_10127_times__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(pos,M)),aa_num_int(pos,N)) = aa_num_int(pos,aa_num_num(times_times_num(M),N)) ).

% times_int_code(3)
tff(fact_10128_minus__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(pos,M)),aa_num_int(pos,N)) = aa_num_int(sub(M),N) ).

% minus_int_code(3)
tff(fact_10129_minus__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(neg,M)),aa_num_int(pos,N)) = aa_num_int(neg,aa_num_num(plus_plus_num(M),N)) ).

% minus_int_code(5)
tff(fact_10130_minus__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(pos,M)),aa_num_int(neg,N)) = aa_num_int(pos,aa_num_num(plus_plus_num(M),N)) ).

% minus_int_code(4)
tff(fact_10131_times__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(pos,M)),aa_num_int(neg,N)) = aa_num_int(neg,aa_num_num(times_times_num(M),N)) ).

% times_int_code(4)
tff(fact_10132_times__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(neg,M)),aa_num_int(pos,N)) = aa_num_int(neg,aa_num_num(times_times_num(M),N)) ).

% times_int_code(5)
tff(fact_10133_times__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(neg,M)),aa_num_int(neg,N)) = aa_num_int(pos,aa_num_num(times_times_num(M),N)) ).

% times_int_code(6)
tff(fact_10134_plus__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(pos,M)),aa_num_int(neg,N)) = aa_num_int(sub(M),N) ).

% plus_int_code(4)
tff(fact_10135_plus__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(neg,M)),aa_num_int(pos,N)) = aa_num_int(sub(N),M) ).

% plus_int_code(5)
tff(fact_10136_Quotient3__real,axiom,
    quotie8981163687542601860t_real(realrel,real2,rep_real) ).

% Quotient3_real
tff(fact_10137_sum__encode__def,axiom,
    ! [X: sum_sum_nat_nat] : aa_Sum8073061460641146765at_nat(nat_sum_encode,X) = sum_ca6763686470577984908at_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aTP_Lamm_py,X) ).

% sum_encode_def
tff(fact_10138_inj__sum__encode,axiom,
    ! [A2: set_Sum_sum_nat_nat] : inj_on6343450744447823682at_nat(nat_sum_encode,A2) ).

% inj_sum_encode
tff(fact_10139_bij__sum__encode,axiom,
    pp(aa_set_nat_bool(bij_be5432664580149595207at_nat(nat_sum_encode,top_to6661820994512907621at_nat),top_top_set_nat)) ).

% bij_sum_encode
tff(fact_10140_sum__encode__eq,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
      ( ( aa_Sum8073061460641146765at_nat(nat_sum_encode,X) = aa_Sum8073061460641146765at_nat(nat_sum_encode,Y) )
    <=> ( X = Y ) ) ).

% sum_encode_eq
tff(fact_10141_surj__sum__encode,axiom,
    image_1320371278474632150at_nat(nat_sum_encode,top_to6661820994512907621at_nat) = top_top_set_nat ).

% surj_sum_encode
tff(fact_10142_int__encode__def,axiom,
    ! [I: int] : aa_int_nat(nat_int_encode,I) = aa_Sum8073061460641146765at_nat(nat_sum_encode,if_Sum_sum_nat_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I),sum_Inl_nat_nat(aa_int_nat(nat2,I)),sum_Inr_nat_nat(aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,I)),one_one_int))))) ).

% int_encode_def
tff(fact_10143_sum__decode__def,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat7431386967378518541at_nat(nat_sum_decode,N) = sum_Inl_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat7431386967378518541at_nat(nat_sum_decode,N) = sum_Inr_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% sum_decode_def
tff(fact_10144_sum__encode__inverse,axiom,
    ! [X: sum_sum_nat_nat] : aa_nat7431386967378518541at_nat(nat_sum_decode,aa_Sum8073061460641146765at_nat(nat_sum_encode,X)) = X ).

% sum_encode_inverse
tff(fact_10145_sum__decode__inverse,axiom,
    ! [N: nat] : aa_Sum8073061460641146765at_nat(nat_sum_encode,aa_nat7431386967378518541at_nat(nat_sum_decode,N)) = N ).

% sum_decode_inverse
tff(fact_10146_surj__sum__decode,axiom,
    image_678696785212003926at_nat(nat_sum_decode,top_top_set_nat) = top_to6661820994512907621at_nat ).

% surj_sum_decode
tff(fact_10147_bij__sum__decode,axiom,
    bij_be4790990086886966983at_nat(nat_sum_decode,top_top_set_nat,top_to6661820994512907621at_nat) ).

% bij_sum_decode
tff(fact_10148_inj__sum__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on5701776251185195458at_nat(nat_sum_decode),A2)) ).

% inj_sum_decode
tff(fact_10149_sum__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,X) = aa_nat7431386967378518541at_nat(nat_sum_decode,Y) )
    <=> ( X = Y ) ) ).

% sum_decode_eq
tff(fact_10150_nth__item_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nth_item_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nth_item_rel),zero_zero_nat))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(accp_nat(nth_item_rel),aa_nat_nat(suc,N2)))
             => ( ! [A6: nat,Aa: nat] :
                    ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N2) = sum_Inl_nat_nat(A6) )
                   => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,A6) = sum_Inl_nat_nat(Aa) )
                     => pp(aa_nat_bool(P,Aa)) ) )
               => ( ! [A6: nat,B9: nat] :
                      ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N2) = sum_Inl_nat_nat(A6) )
                     => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,A6) = sum_Inr_nat_nat(B9) )
                       => pp(aa_nat_bool(P,B9)) ) )
                 => ( ! [B9: nat,Ba: nat,X4: nat,Y4: nat] :
                        ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N2) = sum_Inr_nat_nat(B9) )
                       => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,B9) = sum_Inr_nat_nat(Ba) )
                         => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,Ba) )
                           => pp(aa_nat_bool(P,X4)) ) ) )
                   => ( ! [B9: nat,Ba: nat,X4: nat,Y4: nat] :
                          ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N2) = sum_Inr_nat_nat(B9) )
                         => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,B9) = sum_Inr_nat_nat(Ba) )
                           => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,Ba) )
                             => pp(aa_nat_bool(P,Y4)) ) ) )
                     => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ) )
         => pp(aa_nat_bool(P,A0)) ) ) ) ).

% nth_item.pinduct
tff(fact_10151_int__decode__def,axiom,
    ! [N: nat] : aa_nat_int(nat_int_decode,N) = sum_ca7763040182479039464nt_nat(semiri1314217659103216013at_int,aTP_Lamm_pz,aa_nat7431386967378518541at_nat(nat_sum_decode,N)) ).

% int_decode_def
tff(fact_10152_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_ot,Uu) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(Uu)),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(Uu)),product_fst_int_int(Uu))) ).

% ATP.lambda_1
tff(fact_10153_ATP_Olambda__2,axiom,
    ! [Uu: fun_nat_rat] : aa_fun765758790203053448at_rat(aTP_Lamm_oj,Uu) = if_fun_nat_rat(vanishes(Uu),aTP_Lamm_oe,aTP_Lamm_oc(Uu)) ).

% ATP.lambda_2
tff(fact_10154_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_dy,Uu) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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_3
tff(fact_10155_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_hg,Uu) = aa_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_4
tff(fact_10156_ATP_Olambda__5,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_ox,Uu) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,product_fst_int_int(Uu))),product_snd_int_int(Uu)) ).

% ATP.lambda_5
tff(fact_10157_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_pz,Uu) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int) ).

% ATP.lambda_6
tff(fact_10158_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_iz,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(power_power_real(zero_zero_real),Uu)) ).

% ATP.lambda_7
tff(fact_10159_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_nc,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_8
tff(fact_10160_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_mc,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_10161_ATP_Olambda__10,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ok,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_10
tff(fact_10162_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_om,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_11
tff(fact_10163_ATP_Olambda__12,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_oq,Uu))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu)))) ) ).

% ATP.lambda_12
tff(fact_10164_ATP_Olambda__13,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_pw,Uu) = produc5538323210962509403atural(next,aTP_Lamm_pv(Uu)) ).

% ATP.lambda_13
tff(fact_10165_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ne,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_14
tff(fact_10166_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_py,Uu) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)) ).

% ATP.lambda_15
tff(fact_10167_ATP_Olambda__16,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mv,Uu) = suminf_real(aTP_Lamm_by(Uu)) ).

% ATP.lambda_16
tff(fact_10168_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_oa,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,Uu)) ).

% ATP.lambda_17
tff(fact_10169_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ng,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_18
tff(fact_10170_ATP_Olambda__19,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_lp,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_19
tff(fact_10171_ATP_Olambda__20,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_kp,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_20
tff(fact_10172_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_js,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_21
tff(fact_10173_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa_nat_Extended_enat(aTP_Lamm_pq,Uu) = extended_enat2(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_22
tff(fact_10174_ATP_Olambda__23,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_lm,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_23
tff(fact_10175_ATP_Olambda__24,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_oi,Uu))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N4))) ) ) ) ).

% ATP.lambda_24
tff(fact_10176_ATP_Olambda__25,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_md,Uu))
    <=> ? [I3: int,N4: nat] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I3)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_25
tff(fact_10177_ATP_Olambda__26,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_me,Uu))
    <=> ? [I3: int,J3: int] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I3)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_26
tff(fact_10178_ATP_Olambda__27,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hl(Uu),Uua) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(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_27
tff(fact_10179_ATP_Olambda__28,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_kx,Uu),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu))),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_Code_integer_int(code_int_of_integer,Uu))),one_one_int)) ).

% ATP.lambda_28
tff(fact_10180_ATP_Olambda__29,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_lk,Uu),Uua) = aa_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_num_num(plus_plus_num(aa_Code_integer_num(code_num_of_integer,Uu)),aa_Code_integer_num(code_num_of_integer,Uu))),aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(aa_Code_integer_num(code_num_of_integer,Uu)),aa_Code_integer_num(code_num_of_integer,Uu))),one)) ).

% ATP.lambda_29
tff(fact_10181_ATP_Olambda__30,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_kw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),one_one_nat)) ).

% ATP.lambda_30
tff(fact_10182_ATP_Olambda__31,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kk,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,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,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_31
tff(fact_10183_ATP_Olambda__32,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ow,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uua)) ).

% ATP.lambda_32
tff(fact_10184_ATP_Olambda__33,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_pp(Uu),Uua) = extend3600170679010898289d_enat(aTP_Lamm_po(Uua),if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),Uu) ).

% ATP.lambda_33
tff(fact_10185_ATP_Olambda__34,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_pn(Uu),Uua) = extend3600170679010898289d_enat(aTP_Lamm_pm(Uua),extend5688581933313929465d_enat,Uu) ).

% ATP.lambda_34
tff(fact_10186_ATP_Olambda__35,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_pl(Uu),Uua) = extend3600170679010898289d_enat(aTP_Lamm_pk(Uua),zero_z5237406670263579293d_enat,Uu) ).

% ATP.lambda_35
tff(fact_10187_ATP_Olambda__36,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_pf(Uu),Uua))
    <=> extend8126323780198025762t_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),fTrue,Uu) ) ).

% ATP.lambda_36
tff(fact_10188_ATP_Olambda__37,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_os,Uu),Uua) = 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,product_fst_int_int(Uu)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_37
tff(fact_10189_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bx(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_38
tff(fact_10190_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_fc(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_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(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_39
tff(fact_10191_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_hx(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_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(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_40
tff(fact_10192_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ju,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_41
tff(fact_10193_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jt,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_42
tff(fact_10194_ATP_Olambda__43,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_jp(Uu),Uua))
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).

% ATP.lambda_43
tff(fact_10195_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kg(Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_44
tff(fact_10196_ATP_Olambda__45,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kh(Uu),Uua))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(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_45
tff(fact_10197_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_by(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(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(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_46
tff(fact_10198_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_mw(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_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_47
tff(fact_10199_ATP_Olambda__48,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_48
tff(fact_10200_ATP_Olambda__49,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_oy(Uu),Uua))
    <=> ( ( Uu = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Uua)))
        & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua))) ) ) ).

% ATP.lambda_49
tff(fact_10201_ATP_Olambda__50,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_hy(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_50
tff(fact_10202_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ia(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(Uu),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_51
tff(fact_10203_ATP_Olambda__52,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hz(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_52
tff(fact_10204_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ou,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_53
tff(fact_10205_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_jd(Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_54
tff(fact_10206_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_pb(Uu),Uua) = aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_55
tff(fact_10207_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_56
tff(fact_10208_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh,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_57
tff(fact_10209_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jg,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_58
tff(fact_10210_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_bh(Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_59
tff(fact_10211_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_bi(Uu),Uua))
    <=> ( aa_nat_real(power_power_real(Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_60
tff(fact_10212_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nl(Uu),Uua) = aa_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_61
tff(fact_10213_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_nu(Uu),Uua) = aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_62
tff(fact_10214_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_nt(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_63
tff(fact_10215_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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(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_64
tff(fact_10216_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_do(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_65
tff(fact_10217_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dp(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_66
tff(fact_10218_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_dq(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_67
tff(fact_10219_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_dr(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_68
tff(fact_10220_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_69
tff(fact_10221_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hh(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_70
tff(fact_10222_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hj(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_71
tff(fact_10223_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hi(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_72
tff(fact_10224_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gl(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_73
tff(fact_10225_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gj(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_74
tff(fact_10226_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gk(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_75
tff(fact_10227_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_cp(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_76
tff(fact_10228_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cq(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_77
tff(fact_10229_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_es(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_78
tff(fact_10230_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hf(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_79
tff(fact_10231_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hd(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_80
tff(fact_10232_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_he(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_81
tff(fact_10233_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hc(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_82
tff(fact_10234_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ha(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_83
tff(fact_10235_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hb(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_84
tff(fact_10236_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_na(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_mz(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_85
tff(fact_10237_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nq(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_mz(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_86
tff(fact_10238_ATP_Olambda__87,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_87
tff(fact_10239_ATP_Olambda__88,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ez(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_88
tff(fact_10240_ATP_Olambda__89,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_fa(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_89
tff(fact_10241_ATP_Olambda__90,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ic(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_90
tff(fact_10242_ATP_Olambda__91,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ib(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_91
tff(fact_10243_ATP_Olambda__92,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_lo(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(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_92
tff(fact_10244_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_mh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_93
tff(fact_10245_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_iy(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_94
tff(fact_10246_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ix(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_95
tff(fact_10247_ATP_Olambda__96,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_ol,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_96
tff(fact_10248_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_hn(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_97
tff(fact_10249_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aTP_Lamm_hp(Uu),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_98
tff(fact_10250_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ho(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_99
tff(fact_10251_ATP_Olambda__100,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bu(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_100
tff(fact_10252_ATP_Olambda__101,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bv(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_101
tff(fact_10253_ATP_Olambda__102,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_bw(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_102
tff(fact_10254_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jo,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_103
tff(fact_10255_ATP_Olambda__104,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nm(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_104
tff(fact_10256_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_pj(Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa_nat_nat(aa_nat_fun_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)))))) ).

% ATP.lambda_105
tff(fact_10257_ATP_Olambda__106,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_pg,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).

% ATP.lambda_106
tff(fact_10258_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nk(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_107
tff(fact_10259_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nh(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_108
tff(fact_10260_ATP_Olambda__109,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_dt(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_109
tff(fact_10261_ATP_Olambda__110,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ds(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_110
tff(fact_10262_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dv(Uu),Uua) = aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_111
tff(fact_10263_ATP_Olambda__112,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_du(Uu),Uua) = aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_112
tff(fact_10264_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_113
tff(fact_10265_ATP_Olambda__114,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_dx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_114
tff(fact_10266_ATP_Olambda__115,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_df(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_115
tff(fact_10267_ATP_Olambda__116,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_de(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_116
tff(fact_10268_ATP_Olambda__117,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dg(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_117
tff(fact_10269_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_118
tff(fact_10270_ATP_Olambda__119,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_di(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_119
tff(fact_10271_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_ph,Uu),Uua) = cons_nat(Uu,aa_nat_list_nat(nat_list_decode,Uua)) ).

% ATP.lambda_120
tff(fact_10272_ATP_Olambda__121,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_pe,Uu),Uua))
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_121
tff(fact_10273_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ag,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_122
tff(fact_10274_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_123
tff(fact_10275_ATP_Olambda__124,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_af,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_124
tff(fact_10276_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_125
tff(fact_10277_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ll,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_126
tff(fact_10278_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_mx,Uu),Uua) = aa_nat_real(power_power_real(Uua),Uu) ).

% ATP.lambda_127
tff(fact_10279_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_128
tff(fact_10280_ATP_Olambda__129,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_ln(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_129
tff(fact_10281_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_mj(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_130
tff(fact_10282_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aw,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_131
tff(fact_10283_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ax,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_132
tff(fact_10284_ATP_Olambda__133,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ay,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_133
tff(fact_10285_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_la,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_134
tff(fact_10286_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_135
tff(fact_10287_ATP_Olambda__136,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nf(Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_136
tff(fact_10288_ATP_Olambda__137,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_a(Uu),Uua))
    <=> member_complex(Uua,Uu) ) ).

% ATP.lambda_137
tff(fact_10289_ATP_Olambda__138,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu),Uua))
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_138
tff(fact_10290_ATP_Olambda__139,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_139
tff(fact_10291_ATP_Olambda__140,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_140
tff(fact_10292_ATP_Olambda__141,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_10293_ATP_Olambda__142,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_142
tff(fact_10294_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_pt(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_143
tff(fact_10295_ATP_Olambda__144,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jc(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_144
tff(fact_10296_ATP_Olambda__145,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jb(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_145
tff(fact_10297_ATP_Olambda__146,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nw(Uu),Uua))
    <=> pp(aa_real_bool(Uu,aa_real_real(inverse_inverse_real,Uua))) ) ).

% ATP.lambda_146
tff(fact_10298_ATP_Olambda__147,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_my(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_147
tff(fact_10299_ATP_Olambda__148,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nv(Uu),Uua))
    <=> pp(aa_real_bool(Uu,aa_real_real(uminus_uminus_real,Uua))) ) ).

% ATP.lambda_148
tff(fact_10300_ATP_Olambda__149,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cx(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_149
tff(fact_10301_ATP_Olambda__150,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_nr(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_150
tff(fact_10302_ATP_Olambda__151,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_cy(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_151
tff(fact_10303_ATP_Olambda__152,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cr(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_152
tff(fact_10304_ATP_Olambda__153,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cs(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_153
tff(fact_10305_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_oo,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_on(Uu),Uua)) ).

% ATP.lambda_154
tff(fact_10306_ATP_Olambda__155,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_li,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_lh(Uu),Uua)) ).

% ATP.lambda_155
tff(fact_10307_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_lg,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_lf(Uu),Uua)) ).

% ATP.lambda_156
tff(fact_10308_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_le,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ld(Uu),Uua)) ).

% ATP.lambda_157
tff(fact_10309_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_lc,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_lb(Uu),Uua)) ).

% ATP.lambda_158
tff(fact_10310_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_kz,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ky(Uu),Uua)) ).

% ATP.lambda_159
tff(fact_10311_ATP_Olambda__160,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_mn(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_mm(Uu),Uua)) ).

% ATP.lambda_160
tff(fact_10312_ATP_Olambda__161,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_je(Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_161
tff(fact_10313_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bj,Uu),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_162
tff(fact_10314_ATP_Olambda__163,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_pv(Uu),Uua) = produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uua),times_2397367101498566445atural(Uu,numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_163
tff(fact_10315_ATP_Olambda__164,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_bq(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_164
tff(fact_10316_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_bs(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_165
tff(fact_10317_ATP_Olambda__166,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_br(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_166
tff(fact_10318_ATP_Olambda__167,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_bn(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_167
tff(fact_10319_ATP_Olambda__168,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_bk(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_168
tff(fact_10320_ATP_Olambda__169,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_bm(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_169
tff(fact_10321_ATP_Olambda__170,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_bo(Uu),Uua) = times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_170
tff(fact_10322_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bp(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_171
tff(fact_10323_ATP_Olambda__172,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_bl(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_172
tff(fact_10324_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_pa(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_173
tff(fact_10325_ATP_Olambda__174,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nj(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_174
tff(fact_10326_ATP_Olambda__175,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_px(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_175
tff(fact_10327_ATP_Olambda__176,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_po(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_176
tff(fact_10328_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_pk(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_177
tff(fact_10329_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_pm(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)) ).

% ATP.lambda_178
tff(fact_10330_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_179
tff(fact_10331_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lq,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_180
tff(fact_10332_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_181
tff(fact_10333_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ki,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_182
tff(fact_10334_ATP_Olambda__183,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_al(Uu),Uua))
    <=> ~ member_complex(Uua,Uu) ) ).

% ATP.lambda_183
tff(fact_10335_ATP_Olambda__184,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_am(Uu),Uua))
    <=> ~ member_real(Uua,Uu) ) ).

% ATP.lambda_184
tff(fact_10336_ATP_Olambda__185,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_an(Uu),Uua))
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_185
tff(fact_10337_ATP_Olambda__186,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ao(Uu),Uua))
    <=> ~ member_set_nat(Uua,Uu) ) ).

% ATP.lambda_186
tff(fact_10338_ATP_Olambda__187,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ap(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_187
tff(fact_10339_ATP_Olambda__188,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_aq(Uu),Uua))
    <=> ~ member_int(Uua,Uu) ) ).

% ATP.lambda_188
tff(fact_10340_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nd(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_189
tff(fact_10341_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oc(Uu),Uua) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_190
tff(fact_10342_ATP_Olambda__191,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_fe(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_191
tff(fact_10343_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_cb(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_192
tff(fact_10344_ATP_Olambda__193,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ca(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_193
tff(fact_10345_ATP_Olambda__194,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_rat(aTP_Lamm_cc(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_194
tff(fact_10346_ATP_Olambda__195,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cd(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_195
tff(fact_10347_ATP_Olambda__196,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_ce(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_196
tff(fact_10348_ATP_Olambda__197,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_bz(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_197
tff(fact_10349_ATP_Olambda__198,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_fw(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_198
tff(fact_10350_ATP_Olambda__199,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_er(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_199
tff(fact_10351_ATP_Olambda__200,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_eq(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_200
tff(fact_10352_ATP_Olambda__201,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_lu,Uu),Uua) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_201
tff(fact_10353_ATP_Olambda__202,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fx(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_202
tff(fact_10354_ATP_Olambda__203,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_cz(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_203
tff(fact_10355_ATP_Olambda__204,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_da(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_204
tff(fact_10356_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_db(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_205
tff(fact_10357_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_dc(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_206
tff(fact_10358_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_dd(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_10359_ATP_Olambda__208,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_ff(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_10360_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_cg(Uu),Uua) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_10361_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_complex(aTP_Lamm_ci(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_10362_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_cf(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_10363_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_ch(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_10364_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cj(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_10365_ATP_Olambda__214,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_Code_integer(aTP_Lamm_cl(Uu),Uua) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_10366_ATP_Olambda__215,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_complex(aTP_Lamm_cn(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_10367_ATP_Olambda__216,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_ck(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_10368_ATP_Olambda__217,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_cm(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_10369_ATP_Olambda__218,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_co(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_10370_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_gp(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_10371_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_gm(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_10372_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_go(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_10373_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_gn(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_10374_ATP_Olambda__223,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_nz(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_10375_ATP_Olambda__224,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ny(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_10376_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fh(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_10377_ATP_Olambda__226,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fg(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10378_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_np(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_10379_ATP_Olambda__228,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_nat(aTP_Lamm_im(Uu),Uua) = aa_nat_nat(suc,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_10380_ATP_Olambda__229,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_in(Uu),Uua) = aa_nat_nat(suc,aa_list_nat_nat(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_10381_ATP_Olambda__230,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_io(Uu),Uua) = aa_nat_nat(suc,aa_set_nat_nat(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_10382_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_iq(Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_10383_ATP_Olambda__232,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_nat(aTP_Lamm_ip(Uu),Uua) = aa_nat_nat(suc,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_10384_ATP_Olambda__233,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ar(Uu),Uua))
    <=> ~ pp(aa_real_bool(Uu,Uua)) ) ).

% ATP.lambda_233
tff(fact_10385_ATP_Olambda__234,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_as(Uu),Uua))
    <=> ~ pp(aa_list_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_10386_ATP_Olambda__235,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_at(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_235
tff(fact_10387_ATP_Olambda__236,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_au(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_10388_ATP_Olambda__237,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_av(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_10389_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_pr(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,Uu)) ).

% ATP.lambda_238
tff(fact_10390_ATP_Olambda__239,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_op(Uu),Uua))
    <=> ! [X5: real] :
          ( member_real(X5,Uu)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Uua)) ) ) ).

% ATP.lambda_239
tff(fact_10391_ATP_Olambda__240,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_pu(Uu),Uua))
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),Uu))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),X5)) ) ) ).

% ATP.lambda_240
tff(fact_10392_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_oz(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,Uu) ).

% ATP.lambda_241
tff(fact_10393_ATP_Olambda__242,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_oh(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,Uu) ).

% ATP.lambda_242
tff(fact_10394_ATP_Olambda__243,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_rat(aTP_Lamm_og(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,Uu) ).

% ATP.lambda_243
tff(fact_10395_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hm(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(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_244
tff(fact_10396_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ja(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(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_245
tff(fact_10397_ATP_Olambda__246,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_kd(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_246
tff(fact_10398_ATP_Olambda__247,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_jq(Uu),Uua),Uub) = if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(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_247
tff(fact_10399_ATP_Olambda__248,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jr(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(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_248
tff(fact_10400_ATP_Olambda__249,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ee(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_249
tff(fact_10401_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ef(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_250
tff(fact_10402_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_eg(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_251
tff(fact_10403_ATP_Olambda__252,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ku(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_252
tff(fact_10404_ATP_Olambda__253,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_kv(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,abs_abs_Code_integer(Uu)),Uub))) ).

% ATP.lambda_253
tff(fact_10405_ATP_Olambda__254,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_kt(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),Uub))) ).

% ATP.lambda_254
tff(fact_10406_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_gg(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_255
tff(fact_10407_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_real),zero_zero_real)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_256
tff(fact_10408_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_gi(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_int),zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_257
tff(fact_10409_ATP_Olambda__258,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_ks(Uu),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_258
tff(fact_10410_ATP_Olambda__259,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_kr(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_kq(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_259
tff(fact_10411_ATP_Olambda__260,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ko(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_kn(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_260
tff(fact_10412_ATP_Olambda__261,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_km(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_kl(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_261
tff(fact_10413_ATP_Olambda__262,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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(power_power_real(Uua),Uub)) ).

% ATP.lambda_262
tff(fact_10414_ATP_Olambda__263,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_mp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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(power_power_real(Uu),Uub)) ).

% ATP.lambda_263
tff(fact_10415_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ml(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_264
tff(fact_10416_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ly(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_265
tff(fact_10417_ATP_Olambda__266,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_lz(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(groups4561878855575611511st_nat,Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_266
tff(fact_10418_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_lx(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_267
tff(fact_10419_ATP_Olambda__268,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_be(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_268
tff(fact_10420_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_pd(Uu),Uua),Uub))
    <=> ( member_set_nat(Uub,pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_269
tff(fact_10421_ATP_Olambda__270,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ps(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_270
tff(fact_10422_ATP_Olambda__271,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_271
tff(fact_10423_ATP_Olambda__272,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji(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_272
tff(fact_10424_ATP_Olambda__273,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_kc(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_273
tff(fact_10425_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jv(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_274
tff(fact_10426_ATP_Olambda__275,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_kb(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_275
tff(fact_10427_ATP_Olambda__276,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jy(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_276
tff(fact_10428_ATP_Olambda__277,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jx(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_277
tff(fact_10429_ATP_Olambda__278,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jw(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_278
tff(fact_10430_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ov(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).

% ATP.lambda_279
tff(fact_10431_ATP_Olambda__280,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_or(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uu)) ) ) ).

% ATP.lambda_280
tff(fact_10432_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ob(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_281
tff(fact_10433_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_pc(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_282
tff(fact_10434_ATP_Olambda__283,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_iw(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_283
tff(fact_10435_ATP_Olambda__284,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_bg(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_284
tff(fact_10436_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_et(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_285
tff(fact_10437_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ev(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_286
tff(fact_10438_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_no(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_287
tff(fact_10439_ATP_Olambda__288,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(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_288
tff(fact_10440_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_eu(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_289
tff(fact_10441_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ew(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_290
tff(fact_10442_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jl(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_291
tff(fact_10443_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_jz(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_292
tff(fact_10444_ATP_Olambda__293,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ls,Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_293
tff(fact_10445_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ka(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_294
tff(fact_10446_ATP_Olambda__295,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_gq(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_295
tff(fact_10447_ATP_Olambda__296,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_296
tff(fact_10448_ATP_Olambda__297,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gx(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_297
tff(fact_10449_ATP_Olambda__298,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_gr(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_298
tff(fact_10450_ATP_Olambda__299,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_299
tff(fact_10451_ATP_Olambda__300,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gy(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_300
tff(fact_10452_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_gs(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_301
tff(fact_10453_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gz(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_302
tff(fact_10454_ATP_Olambda__303,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_gt(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_303
tff(fact_10455_ATP_Olambda__304,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_304
tff(fact_10456_ATP_Olambda__305,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fu(Uu),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_305
tff(fact_10457_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_306
tff(fact_10458_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_lw(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_307
tff(fact_10459_ATP_Olambda__308,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_308
tff(fact_10460_ATP_Olambda__309,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_id(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_309
tff(fact_10461_ATP_Olambda__310,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ie(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_310
tff(fact_10462_ATP_Olambda__311,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_if(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_311
tff(fact_10463_ATP_Olambda__312,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ih(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_312
tff(fact_10464_ATP_Olambda__313,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ig(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_313
tff(fact_10465_ATP_Olambda__314,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fs(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_314
tff(fact_10466_ATP_Olambda__315,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_315
tff(fact_10467_ATP_Olambda__316,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_lv,Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_316
tff(fact_10468_ATP_Olambda__317,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(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_317
tff(fact_10469_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_en(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_318
tff(fact_10470_ATP_Olambda__319,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ft(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_319
tff(fact_10471_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ej(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_320
tff(fact_10472_ATP_Olambda__321,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(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_321
tff(fact_10473_ATP_Olambda__322,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_322
tff(fact_10474_ATP_Olambda__323,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_mi(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_323
tff(fact_10475_ATP_Olambda__324,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nx(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_324
tff(fact_10476_ATP_Olambda__325,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_mk(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_325
tff(fact_10477_ATP_Olambda__326,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_od(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu,Uub))),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uua,Uub))) ).

% ATP.lambda_326
tff(fact_10478_ATP_Olambda__327,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_hs(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_327
tff(fact_10479_ATP_Olambda__328,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_328
tff(fact_10480_ATP_Olambda__329,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_329
tff(fact_10481_ATP_Olambda__330,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ht(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_330
tff(fact_10482_ATP_Olambda__331,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_331
tff(fact_10483_ATP_Olambda__332,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_332
tff(fact_10484_ATP_Olambda__333,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ni(Uu),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_333
tff(fact_10485_ATP_Olambda__334,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_gb(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_334
tff(fact_10486_ATP_Olambda__335,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ge(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_335
tff(fact_10487_ATP_Olambda__336,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_gc(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_336
tff(fact_10488_ATP_Olambda__337,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_gd(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_337
tff(fact_10489_ATP_Olambda__338,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hw(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_338
tff(fact_10490_ATP_Olambda__339,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fz(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_339
tff(fact_10491_ATP_Olambda__340,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ei(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_340
tff(fact_10492_ATP_Olambda__341,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_dk(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_341
tff(fact_10493_ATP_Olambda__342,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_dj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_342
tff(fact_10494_ATP_Olambda__343,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_dl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_343
tff(fact_10495_ATP_Olambda__344,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_344
tff(fact_10496_ATP_Olambda__345,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_dn(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_345
tff(fact_10497_ATP_Olambda__346,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_pi(Uu),Uua),Uub) = aa_real_real(powr_real2(Uu),Uua) ).

% ATP.lambda_346
tff(fact_10498_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_fb(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_347
tff(fact_10499_ATP_Olambda__348,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_eh(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_348
tff(fact_10500_ATP_Olambda__349,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ns(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_349
tff(fact_10501_ATP_Olambda__350,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ct(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_350
tff(fact_10502_ATP_Olambda__351,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_cu(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_351
tff(fact_10503_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_352
tff(fact_10504_ATP_Olambda__353,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nn(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_353
tff(fact_10505_ATP_Olambda__354,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_jn(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_jm(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_354
tff(fact_10506_ATP_Olambda__355,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_355
tff(fact_10507_ATP_Olambda__356,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ms(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_356
tff(fact_10508_ATP_Olambda__357,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_mq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_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(power_power_real(aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_357
tff(fact_10509_ATP_Olambda__358,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_mr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_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(power_power_real(aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_358
tff(fact_10510_ATP_Olambda__359,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ky(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_359
tff(fact_10511_ATP_Olambda__360,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jk(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_360
tff(fact_10512_ATP_Olambda__361,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jj(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_361
tff(fact_10513_ATP_Olambda__362,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ld(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_362
tff(fact_10514_ATP_Olambda__363,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_kq(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_363
tff(fact_10515_ATP_Olambda__364,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_lb(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_364
tff(fact_10516_ATP_Olambda__365,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_lf(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_365
tff(fact_10517_ATP_Olambda__366,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_lh(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_366
tff(fact_10518_ATP_Olambda__367,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_on(Uu),Uua),Uub),Uuc))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_367
tff(fact_10519_ATP_Olambda__368,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ex(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_368
tff(fact_10520_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_369
tff(fact_10521_ATP_Olambda__370,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_jm(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_370
tff(fact_10522_ATP_Olambda__371,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gf(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_371
tff(fact_10523_ATP_Olambda__372,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_cv(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_372
tff(fact_10524_ATP_Olambda__373,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_cw(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_373
tff(fact_10525_ATP_Olambda__374,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_kn(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_374
tff(fact_10526_ATP_Olambda__375,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_kl(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_375
tff(fact_10527_ATP_Olambda__376,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_mt(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_ms(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(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(divide_divide_real(aa_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_376
tff(fact_10528_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_ma(Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
           => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I3)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I3)) ) )
        & ( ( Uuc = Uud )
         => ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
            & ! [X5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X5))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_377
tff(fact_10529_ATP_Olambda__378,axiom,
    ! [Uu: code_integer,Uua: complex] : aa_com4217951221296830281nteger(aTP_Lamm_fo(Uu),Uua) = Uu ).

% ATP.lambda_378
tff(fact_10530_ATP_Olambda__379,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_fp(Uu),Uua) = Uu ).

% ATP.lambda_379
tff(fact_10531_ATP_Olambda__380,axiom,
    ! [Uu: code_integer,Uua: int] : aa_int_Code_integer(aTP_Lamm_fq(Uu),Uua) = Uu ).

% ATP.lambda_380
tff(fact_10532_ATP_Olambda__381,axiom,
    ! [Uu: real,Uua: complex] : aa_complex_real(aTP_Lamm_fk(Uu),Uua) = Uu ).

% ATP.lambda_381
tff(fact_10533_ATP_Olambda__382,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ep(Uu),Uua) = Uu ).

% ATP.lambda_382
tff(fact_10534_ATP_Olambda__383,axiom,
    ! [Uu: real,Uua: int] : aa_int_real(aTP_Lamm_fl(Uu),Uua) = Uu ).

% ATP.lambda_383
tff(fact_10535_ATP_Olambda__384,axiom,
    ! [Uu: rat,Uua: complex] : aa_complex_rat(aTP_Lamm_fr(Uu),Uua) = Uu ).

% ATP.lambda_384
tff(fact_10536_ATP_Olambda__385,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lt(Uu),Uua) = Uu ).

% ATP.lambda_385
tff(fact_10537_ATP_Olambda__386,axiom,
    ! [Uu: nat,Uua: complex] : aa_complex_nat(aTP_Lamm_fm(Uu),Uua) = Uu ).

% ATP.lambda_386
tff(fact_10538_ATP_Olambda__387,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aTP_Lamm_fn(Uu),Uua) = Uu ).

% ATP.lambda_387
tff(fact_10539_ATP_Olambda__388,axiom,
    ! [Uu: int,Uua: complex] : aa_complex_int(aTP_Lamm_fi(Uu),Uua) = Uu ).

% ATP.lambda_388
tff(fact_10540_ATP_Olambda__389,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_fj(Uu),Uua) = Uu ).

% ATP.lambda_389
tff(fact_10541_ATP_Olambda__390,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_az,Uu) = Uu ).

% ATP.lambda_390
tff(fact_10542_ATP_Olambda__391,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ba,Uu) = Uu ).

% ATP.lambda_391
tff(fact_10543_ATP_Olambda__392,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_bb,Uu) = Uu ).

% ATP.lambda_392
tff(fact_10544_ATP_Olambda__393,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bc,Uu) = Uu ).

% ATP.lambda_393
tff(fact_10545_ATP_Olambda__394,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_bd,Uu) = Uu ).

% ATP.lambda_394
tff(fact_10546_ATP_Olambda__395,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_fd,Uu) = zero_zero_complex ).

% ATP.lambda_395
tff(fact_10547_ATP_Olambda__396,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ah,Uu) = zero_zero_real ).

% ATP.lambda_396
tff(fact_10548_ATP_Olambda__397,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ai,Uu) = zero_zero_rat ).

% ATP.lambda_397
tff(fact_10549_ATP_Olambda__398,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_dz,Uu) = zero_zero_real ).

% ATP.lambda_398
tff(fact_10550_ATP_Olambda__399,axiom,
    ! [Uu: nat] : aa_nat_rat(aTP_Lamm_oe,Uu) = zero_zero_rat ).

% ATP.lambda_399
tff(fact_10551_ATP_Olambda__400,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aj,Uu) = zero_zero_nat ).

% ATP.lambda_400
tff(fact_10552_ATP_Olambda__401,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ea,Uu) = zero_zero_int ).

% ATP.lambda_401
tff(fact_10553_ATP_Olambda__402,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ak,Uu) = zero_zero_int ).

% ATP.lambda_402
tff(fact_10554_ATP_Olambda__403,axiom,
    ! [Uu: complex] : aa_complex_real(aTP_Lamm_ir,Uu) = one_one_real ).

% ATP.lambda_403
tff(fact_10555_ATP_Olambda__404,axiom,
    ! [Uu: complex] : aa_complex_nat(aTP_Lamm_ii,Uu) = one_one_nat ).

% ATP.lambda_404
tff(fact_10556_ATP_Olambda__405,axiom,
    ! [Uu: list_nat] : aa_list_nat_real(aTP_Lamm_is,Uu) = one_one_real ).

% ATP.lambda_405
tff(fact_10557_ATP_Olambda__406,axiom,
    ! [Uu: list_nat] : aa_list_nat_nat(aTP_Lamm_ij,Uu) = one_one_nat ).

% ATP.lambda_406
tff(fact_10558_ATP_Olambda__407,axiom,
    ! [Uu: set_nat] : aa_set_nat_real(aTP_Lamm_it,Uu) = one_one_real ).

% ATP.lambda_407
tff(fact_10559_ATP_Olambda__408,axiom,
    ! [Uu: set_nat] : aa_set_nat_nat(aTP_Lamm_ik,Uu) = one_one_nat ).

% ATP.lambda_408
tff(fact_10560_ATP_Olambda__409,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_iv,Uu) = one_one_real ).

% ATP.lambda_409
tff(fact_10561_ATP_Olambda__410,axiom,
    ! [Uu: nat] : aa_nat_rat(aTP_Lamm_of,Uu) = one_one_rat ).

% ATP.lambda_410
tff(fact_10562_ATP_Olambda__411,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_eb,Uu) = one_one_nat ).

% ATP.lambda_411
tff(fact_10563_ATP_Olambda__412,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ec,Uu) = one_one_int ).

% ATP.lambda_412
tff(fact_10564_ATP_Olambda__413,axiom,
    ! [Uu: int] : aa_int_real(aTP_Lamm_iu,Uu) = one_one_real ).

% ATP.lambda_413
tff(fact_10565_ATP_Olambda__414,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_il,Uu) = one_one_nat ).

% ATP.lambda_414
tff(fact_10566_ATP_Olambda__415,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ed,Uu) = one_one_int ).

% ATP.lambda_415
tff(fact_10567_ATP_Olambda__416,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ke,Uu))
    <=> $false ) ).

% ATP.lambda_416
tff(fact_10568_ATP_Olambda__417,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_kf,Uu))
    <=> $true ) ).

% ATP.lambda_417

% Helper facts (66)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(if_num(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(if_num(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fTrue,X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(if_real(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(if_real(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(fequal_num,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(fequal_num,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(fequal_complex(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(fequal_complex(X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : if_Extended_enat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : if_Extended_enat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      | pp(fequal_Extended_enat(X,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(fequal_Extended_enat(X,Y))
      | ( X = Y ) ) ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,X)) ) ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] : if_Sum_sum_nat_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] : if_Sum_sum_nat_nat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R),aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m)) ).

%------------------------------------------------------------------------------