TPTP Problem File: ITP231

TPTP Problem File: ITP231_1.p

%------------------------------------------------------------------------------
% File     : ITP231_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00200_011877
% 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    : 0067_VEBT_InsertCorrectness_00200_011877 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 4425 (1420 unt;1110 typ;   0 def)
%            Number of atoms       : 8645 (3535 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 6284 ( 954   ~; 174   |; 690   &)
%                                         ( 726 <=>;3740  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of types       :  194 ( 193 usr)
%            Number of type conns  : 1098 ( 690   >; 408   *;   0   +;   0  <<)
%            Number of predicates  :   55 (  52 usr;   2 prp; 0-4 aty)
%            Number of functors    :  865 ( 865 usr; 227 con; 0-8 aty)
%            Number of variables   : 7936 (7601   !; 335   ?;7936   :)
% 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 20:37:09.990
%------------------------------------------------------------------------------
% Could-be-implicit typings (193)
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    aTP_Lamm_ax: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: nat > fun_co1378794544173401837x_bool ).

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

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_re4238438383274954274t_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: nat > fun_na3244735409550470327at_nat ).

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

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_real_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_real_bool ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_bool > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: num > fun_Co2470855966376689427nteger ).

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

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_re4238438383274954274t_real > fun_real_real ).

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

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_real_real ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_real_real ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_real > fun_fu6119738925743952495t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_real_bool > fun_re1255401033344146673l_bool ).

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

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_real_bool ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_real_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: nat > fun_na5614638261385347865t_bool ).

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

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_Pr803113274115125473t_bool ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: real > int ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

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

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun_Pr5837708615769641901m_bool ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun_Pr5837708615769641901m_bool ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: ( int * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: ( nat * int ) > int ).

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tff(sy_c_Code__Numeral_Obit__cut__integer,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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tff(sy_c_Code__Numeral_Ointeger__of__num,type,
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tff(sy_c_Code__Numeral_Onat__of__integer,type,
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tff(sy_c_Code__Numeral_Onegative,type,
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tff(sy_c_Code__Target__Int_Onegative,type,
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tff(sy_c_Complex_OArg,type,
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tff(sy_c_Complex_Ocis,type,
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tff(sy_c_Complex_Ocnj,type,
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tff(sy_c_Complex_Ocomplex_OComplex,type,
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tff(sy_c_Complex_Ocomplex_OIm,type,
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tff(sy_c_Complex_Ocomplex_ORe,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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tff(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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tff(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
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tff(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: ( num * num ) > num ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
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    nat_list_encode_rel: fun_li8598321315282354825t_bool ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr248683899324676269t_bool ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: ( num * num ) > int ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: fun_num_Code_integer ).

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

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

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Num_Osqr,type,
    sqr: num > num ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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

tff(sy_c_Option_Ooption_Ocase__option_001t__HOL__Obool_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o7912701867834613557at_nat: ( bool * fun_Pr7757163693017731481t_bool * option4927543243414619207at_nat ) > $o ).

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

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

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

tff(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
    map_option_num_num: ( fun_num_num * option_num ) > option_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: ( code_integer * code_integer ) > bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > fun_Ex8563670519792538868t_bool ).

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: num > fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: ( rat * rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: ( code_integer * code_integer ) > bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_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: num > fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: ( rat * rat ) > $o ).

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 * set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: ( set_VEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: ( int * int ) > int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

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

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

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

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

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
    top_top_set_bool: set_bool ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: ( complex * 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: nat > fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: ( rat * nat ) > rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
    produc5135875973608410785r_bool: ( code_integer * bool ) > produc7417142470700560111r_bool ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: ( int * int ) > product_prod_int_int ).

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

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

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

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

tff(sy_c_Product__Type_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__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 * product_prod_nat_nat ) > fun_Pr7757163693017731481t_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 * product_prod_nat_nat ) > fun_Pr2118491707810114563at_nat ).

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

tff(sy_c_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: product_prod_nat_nat > 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: product_prod_nat_nat > nat ).

tff(sy_c_Rat_OFract,type,
    fract: ( int * 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_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

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_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

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

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

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

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: ( real * complex ) > complex ).

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

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

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: ( int * int ) > int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: nat > fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: ( rat * 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: ( code_integer * code_integer ) > 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_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: ( int * int ) > int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: 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__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: bool > nat ).

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

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

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_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_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

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

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

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( fun_nat_real * set_nat ) > set_real ).

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__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_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

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

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

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

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 * nat ) > set_nat ).

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_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 * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: 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__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: 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 * 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 * nat ) > set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: 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__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: 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_Oascii__of,type,
    ascii_of: char > char ).

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

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

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

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

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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_Ohigh,type,
    vEBT_VEBT_high: 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: 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__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * 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__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 > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( fun_Pr5837708615769641901m_bool * product_prod_num_num ) > $o ).

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_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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

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

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    aa_Cod2825494507867081354r_bool: ( fun_Co7333321365496032713r_bool * code_integer ) > produc7417142470700560111r_bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod4161879620478718930nteger: ( fun_Co1981900195452921617nteger * code_integer ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    aa_Cod1242632294114111009er_int: ( fun_Co1092983284983208416er_int * code_integer ) > fun_Code_integer_int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
    aa_Cod6128727250572145423er_num: ( fun_Co8521658056209057742er_num * code_integer ) > fun_Code_integer_num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod8380929501330563836nteger: ( fun_Co2470855966376689427nteger * code_integer ) > fun_Co6597186368970283863nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J_J,type,
    aa_Cod4981436349852597092r_bool: ( fun_Co3025358997185726883r_bool * code_integer ) > fun_Co7333321365496032713r_bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    aa_complex_complex: ( fun_complex_complex * complex ) > complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__HOL__Obool,type,
    aa_complex_bool: ( fun_complex_bool * complex ) > bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    aa_com1553036987509291606x_bool: ( fun_co1378794544173401837x_bool * complex ) > fun_complex_bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
    aa_Ext7969882492699152700d_enat: ( fun_Ex5852502975177633731d_enat * extended_enat ) > extended_enat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__HOL__Obool,type,
    aa_Ext2772979874139974381t_bool: ( fun_Ex8563670519792538868t_bool * extended_enat ) > bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Extended____Nat__Oenat_Mt__Extended____Nat__Oenat_J,type,
    aa_Ext3475206891998675565d_enat: ( fun_Ex8341929632020906428d_enat * extended_enat ) > fun_Ex5852502975177633731d_enat ).

tff(sy_c_aa_001t__HOL__Obool_001t__HOL__Obool,type,
    aa_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_aa_001t__HOL__Obool_001t__Int__Oint,type,
    aa_bool_int: ( fun_bool_int * bool ) > int ).

tff(sy_c_aa_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    aa_boo8970175072276269406l_bool: ( fun_bo1841576699660343469l_bool * bool ) > fun_bool_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__HOL__Obool,type,
    aa_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_aa_001t__Int__Oint_001t__Int__Oint,type,
    aa_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_aa_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_int7462565939943104719nt_int: ( fun_in4211310926729683942nt_int * int ) > product_prod_int_int ).

tff(sy_c_aa_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Int__Oint_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Int__Oint_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,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__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__HOL__Obool_J_J,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__HOL__Obool,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Int__Oint,type,
    aa_nat_int: ( fun_nat_int * nat ) > int ).

tff(sy_c_aa_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_nat_option_num: ( fun_nat_option_num * nat ) > option_num ).

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tff(sy_c_aa_001t__Nat__Onat_001t__Real__Oreal,type,
    aa_nat_real: ( fun_nat_real * nat ) > real ).

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 ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_nat_fun_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_nat8803788312118491388at_nat: ( fun_na4138050610385705875at_nat * nat ) > fun_na1476630412504255114at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    aa_nat8637530788439456283t_bool: ( fun_na3408937105440547634t_bool * nat ) > fun_nat_fun_nat_bool ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_nat3783148529270815480at_nat: ( fun_na3244735409550470327at_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_nat4158579785191972357at_nat: ( fun_na4143055713577223196at_nat * nat ) > fun_na4138050610385705875at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_J,type,
    aa_nat4909001760120733684t_bool: ( fun_na4786964484885125043t_bool * nat ) > fun_na4827981024643048934t_bool ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    aa_nat5682243440470347102at_nat: ( fun_na1668100129766719005at_nat * nat ) > fun_na8713714802964096592at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_nat3713591252005393295on_num: ( fun_na7595719129879163854on_num * nat ) > fun_num_option_num ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
    aa_nat4401263874371387303t_bool: ( fun_na4827981024643048934t_bool * nat ) > fun_Pr7757163693017731481t_bool ).

tff(sy_c_aa_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,
    aa_nat7997280740862404369at_nat: ( fun_na8713714802964096592at_nat * nat ) > fun_Pr2118491707810114563at_nat ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_nat_fun_real_real: ( fun_na6724076748692969250l_real * nat ) > fun_real_real ).

tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    aa_nat1308781354268537904t_real: ( fun_na5225841814499866095t_real * nat ) > fun_re4238438383274954274t_real ).

tff(sy_c_aa_001t__Num__Onum_001t__Code____Numeral__Ointeger,type,
    aa_num_Code_integer: ( fun_num_Code_integer * num ) > code_integer ).

tff(sy_c_aa_001t__Num__Onum_001t__HOL__Obool,type,
    aa_num_bool: ( fun_num_bool * num ) > bool ).

tff(sy_c_aa_001t__Num__Onum_001t__Int__Oint,type,
    aa_num_int: ( fun_num_int * num ) > int ).

tff(sy_c_aa_001t__Num__Onum_001t__Num__Onum,type,
    aa_num_num: ( fun_num_num * num ) > num ).

tff(sy_c_aa_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    aa_num_option_num: ( fun_num_option_num * num ) > option_num ).

tff(sy_c_aa_001t__Num__Onum_001t__fun_It__Num__Onum_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    aa_num6390622462783740633on_num: ( fun_nu1049378303802735384on_num * num ) > fun_num_option_num ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Pro4693618923521437820nteger: ( fun_Pr5876161306168176259nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__HOL__Obool,type,
    aa_Pro4501881432599481050t_bool: ( fun_Pr803113274115125473t_bool * product_prod_int_int ) > bool ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__HOL__Obool,type,
    aa_Pro2232559814647311250t_bool: ( fun_Pr7757163693017731481t_bool * product_prod_nat_nat ) > bool ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    aa_Pro4243173770044397973at_nat: ( fun_Pr2602117396471556196at_nat * product_prod_nat_nat ) > nat ).

tff(sy_c_aa_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_Pro262675715343573244at_nat: ( fun_Pr2118491707810114563at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_aa_001t__Rat__Orat_001t__HOL__Obool,type,
    aa_rat_bool: ( fun_rat_bool * rat ) > bool ).

tff(sy_c_aa_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_rat7754553326175492643nt_int: ( fun_ra4503298312962071866nt_int * rat ) > product_prod_int_int ).

tff(sy_c_aa_001t__Real__Oreal_001t__HOL__Obool,type,
    aa_real_bool: ( fun_real_bool * real ) > bool ).

tff(sy_c_aa_001t__Real__Oreal_001t__Real__Oreal,type,
    aa_real_real: ( fun_real_real * real ) > real ).

tff(sy_c_aa_001t__Real__Oreal_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_real_fun_nat_real: ( fun_re4238438383274954274t_real * real ) > fun_nat_real ).

tff(sy_c_aa_001t__Real__Oreal_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    aa_rea1103652080130114956l_real: ( fun_re4112884445705685835l_real * real ) > fun_na6724076748692969250l_real ).

tff(sy_c_aa_001t__Real__Oreal_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_rea7809428808904475482l_bool: ( fun_re1255401033344146673l_bool * real ) > fun_real_bool ).

tff(sy_c_aa_001t__Real__Oreal_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_rea7340605660300375871l_real: ( fun_re7681423839302404990l_real * real ) > fun_real_real ).

tff(sy_c_aa_001t__Real__Oreal_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    aa_rea7841385751566875788t_real: ( fun_re8542384036533109323t_real * real ) > fun_re4238438383274954274t_real ).

tff(sy_c_aa_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    aa_set_nat_nat: ( fun_set_nat_nat * set_nat ) > nat ).

tff(sy_c_aa_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool,type,
    aa_set8260762808118238015T_bool: ( fun_se448528567104302406T_bool * set_VEBT_VEBT ) > bool ).

tff(sy_c_aa_001t__String__Ochar_001t__Nat__Onat,type,
    aa_char_nat: ( fun_char_nat * char ) > nat ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    aa_VEBT_VEBT_bool: ( fun_VEBT_VEBT_bool * vEBT_VEBT ) > bool ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    aa_VEBT_VEBT_nat: ( fun_VEBT_VEBT_nat * vEBT_VEBT ) > nat ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    aa_VEB8047906998784800507l_bool: ( fun_VE2066785035971159626l_bool * vEBT_VEBT ) > fun_bool_bool ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    aa_VEB4832260192069861672t_bool: ( fun_VE3763956218935105199t_bool * vEBT_VEBT ) > fun_nat_bool ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    aa_VEB1917483692448707121t_bool: ( fun_VE491066615515474744t_bool * vEBT_VEBT ) > fun_nat_fun_nat_bool ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Set__Oset_It__VEBT____Definitions__OVEBT_J_Mt__HOL__Obool_J,type,
    aa_VEB289770264548726718T_bool: ( fun_VE7346385682144900493T_bool * vEBT_VEBT ) > fun_se448528567104302406T_bool ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_fun2509385170507305080nteger: ( fun_fu4800041728187187127nteger * fun_Co8499238673658658103nteger ) > fun_Pr5876161306168176259nteger ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool,type,
    aa_fun_nat_bool_bool: ( fun_fu7168253649841883962l_bool * fun_nat_bool ) > bool ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Int__Oint_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun4551936527957760560t_real: ( fun_fu6119738925743952495t_real * fun_nat_int ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun5791449513213591928at_nat: ( fun_fu7002938529518989751at_nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun1538629612289296340t_real: ( fun_fu3106432010075488275t_real * fun_nat_nat ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6650658898577385413at_nat: ( fun_fu4413992040155564420at_nat * fun_nat_nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun1839206346889086768t_real: ( fun_fu6899955116716589935t_real * fun_nat_real ) > fun_nat_real ).

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

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__HOL__Obool,type,
    aa_fun3352386287069783311l_bool: ( fun_fu7511176317168184086l_bool * fun_real_bool ) > bool ).

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

tff(sy_c_fAll_001t__VEBT____Definitions__OVEBT,type,
    fAll_VEBT_VEBT: fun_VEBT_VEBT_bool > bool ).

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

tff(sy_c_fEx_001t__Nat__Onat,type,
    fEx_nat: fun_fu7168253649841883962l_bool ).

tff(sy_c_fEx_001t__Real__Oreal,type,
    fEx_real: fun_fu7511176317168184086l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: bool > fun_bool_bool ).

tff(sy_c_fdisj,type,
    fdisj: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
    fequal_Code_integer: ( code_integer * code_integer ) > bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1841576699660343469l_bool ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: ( complex * set_complex ) > $o ).

tff(sy_c_member_001t__HOL__Obool,type,
    member_bool: ( bool * set_bool ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_int ) > $o ).

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

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: ( nat * set_nat ) > $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__Real__Oreal,type,
    member_real: ( real * set_real ) > $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_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (3264)
tff(fact_0_abcdef,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),xa)) ).

% abcdef
tff(fact_1__C4_Ohyps_C_I7_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "4.hyps"(7)
tff(fact_2_False,axiom,
    ~ ( ( xa = mi )
      | ( xa = ma ) ) ).

% False
tff(fact_3__092_060open_062mi_A_092_060noteq_062_Amax_Ama_Ax_092_060close_062,axiom,
    mi != aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa) ).

% \<open>mi \<noteq> max ma x\<close>
tff(fact_4__C8_C,axiom,
    na = m ).

% "8"
tff(fact_5_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(vEBT_VEBT_high(X),D),aa_nat_nat(vEBT_VEBT_low(X),D)),D) = X ).

% bit_split_inv
tff(fact_6_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
tff(fact_7_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(plus_plus_nat(N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Ma),N)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M))) ) ).

% high_bound_aux
tff(fact_8__C3_C,axiom,
    deg = aa_nat_nat(plus_plus_nat(na),m) ).

% "3"
tff(fact_9__C4_Oprems_C,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),deg))) ).

% "4.prems"
tff(fact_10__C7_C,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m)))
       => ( ( ( aa_nat_nat(vEBT_VEBT_high(ma),na) = I )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I)),aa_nat_nat(vEBT_VEBT_low(ma),na))) )
          & ! [Y: nat] :
              ( ( ( aa_nat_nat(vEBT_VEBT_high(Y),na) = I )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I)),aa_nat_nat(vEBT_VEBT_low(Y),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),Y))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),ma)) ) ) ) ) ) ).

% "7"
tff(fact_11__C4_Ohyps_C_I8_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),deg))) ).

% "4.hyps"(8)
tff(fact_12_highlowprop,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_low(xa),na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),na))) ) ).

% highlowprop
tff(fact_13__C4_C,axiom,
    ! [I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m)))
     => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I)),X_1))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I)) ) ) ).

% "4"
tff(fact_14__C5_C,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ).

% "5"
tff(fact_15__C2_C,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m) ).

% "2"
tff(fact_16__C15_C,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),deg))) ) ).

% "15"
tff(fact_17_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(X),N))),aa_nat_nat(vEBT_VEBT_low(X),N))) ) ).

% in_children_def
tff(fact_18__C6_C,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),deg))) ) ).

% "6"
tff(fact_19__C14_C,axiom,
    ( ( mi = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa) )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(append_VEBT_VEBT(take_VEBT_VEBT(aa_nat_nat(vEBT_VEBT_high(xa),na),treeList),append_VEBT_VEBT(cons_VEBT_VEBT(vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(vEBT_VEBT_low(xa),na)),nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(aa_nat_nat(vEBT_VEBT_high(xa),na)),one_one_nat),treeList))))))
       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ).

% "14"
tff(fact_20_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y2: 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(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y2)) ) ) ).

% power_increasing_iff
tff(fact_21_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y2: nat] :
      ( ord_less_rat(one_one_rat,B)
     => ( ord_less_eq_rat(power_power_rat(B,X),power_power_rat(B,Y2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y2)) ) ) ).

% power_increasing_iff
tff(fact_22_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y2: 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(power_power_nat(B),X)),aa_nat_nat(power_power_nat(B),Y2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y2)) ) ) ).

% power_increasing_iff
tff(fact_23_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y2: 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),Y2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y2)) ) ) ).

% power_increasing_iff
tff(fact_24_one__add__one,axiom,
    plus_plus_complex(one_one_complex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_25_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_26_one__add__one,axiom,
    plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_27_one__add__one,axiom,
    aa_nat_nat(plus_plus_nat(one_one_nat),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_28_one__add__one,axiom,
    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_29_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y2: 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(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2)) ) ) ).

% power_strict_increasing_iff
tff(fact_30_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y2: nat] :
      ( ord_less_rat(one_one_rat,B)
     => ( ord_less_rat(power_power_rat(B,X),power_power_rat(B,Y2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2)) ) ) ).

% power_strict_increasing_iff
tff(fact_31_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y2: 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(power_power_nat(B),X)),aa_nat_nat(power_power_nat(B),Y2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2)) ) ) ).

% power_strict_increasing_iff
tff(fact_32_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y2: 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),Y2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2)) ) ) ).

% power_strict_increasing_iff
tff(fact_33_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_complex(numera6690914467698888265omplex(N),one_one_complex) = numera6690914467698888265omplex(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_34_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(numeral_numeral_real(N)),one_one_real) = numeral_numeral_real(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_35_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_rat(numeral_numeral_rat(N),one_one_rat) = numeral_numeral_rat(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_36_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(plus_plus_nat(numeral_numeral_nat(N)),one_one_nat) = numeral_numeral_nat(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_37_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_int(aa_num_int(numeral_numeral_int,N),one_one_int) = aa_num_int(numeral_numeral_int,plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_38_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_complex(one_one_complex,numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_39_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(one_one_real),numeral_numeral_real(N)) = numeral_numeral_real(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_40_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(N)) = numeral_numeral_rat(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_41_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(plus_plus_nat(one_one_nat),numeral_numeral_nat(N)) = numeral_numeral_nat(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_42_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_int(one_one_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_43_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_44_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ord_less_rat(one_one_rat,numeral_numeral_rat(N))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_45_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),numeral_numeral_nat(N)))
    <=> pp(aa_num_bool(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_46_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(ord_less_num(one),N)) ) ).

% one_less_numeral_iff
tff(fact_47_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),one_one_real))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_48_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ord_less_eq_rat(numeral_numeral_rat(N),one_one_rat)
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_49_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(N)),one_one_nat))
    <=> pp(aa_num_bool(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_50_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(ord_less_eq_num(N),one)) ) ).

% numeral_le_one_iff
tff(fact_51__C13_C,axiom,
    ! [I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m)))
     => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(append_VEBT_VEBT(take_VEBT_VEBT(aa_nat_nat(vEBT_VEBT_high(xa),na),treeList),append_VEBT_VEBT(cons_VEBT_VEBT(vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(vEBT_VEBT_low(xa),na)),nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(aa_nat_nat(vEBT_VEBT_high(xa),na)),one_one_nat),treeList))),I)),X_1))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na))),vEBT_vebt_insert(summary,aa_nat_nat(vEBT_VEBT_high(xa),na)),summary)),I)) ) ) ).

% "13"
tff(fact_52_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_13)) ) ).

% not_min_Null_member
tff(fact_53_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,N: nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X3,set_Pr5648618587558075414at_nat(Xs))
         => pp(aa_Pro2232559814647311250t_bool(P,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
       => pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,N))) ) ) ).

% inthall
tff(fact_54_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,nth_complex(Xs,N))) ) ) ).

% inthall
tff(fact_55_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_56_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_57_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_58_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X3: nat] :
          ( 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),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_59_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_60_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_61_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_62_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_63_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_nat(M) = numeral_numeral_nat(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_64_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_65_concat__inth,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT,Ys: list_VEBT_VEBT] : nth_VEBT_VEBT(append_VEBT_VEBT(Xs,append_VEBT_VEBT(cons_VEBT_VEBT(X,nil_VEBT_VEBT),Ys)),size_s6755466524823107622T_VEBT(Xs)) = X ).

% concat_inth
tff(fact_66_concat__inth,axiom,
    ! [Xs: list_bool,X: bool,Ys: list_bool] :
      ( pp(aa_nat_bool(nth_bool(append_bool(Xs,append_bool(cons_bool(X,nil_bool),Ys))),size_size_list_bool(Xs)))
    <=> pp(X) ) ).

% concat_inth
tff(fact_67_concat__inth,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] : aa_nat_nat(nth_nat(append_nat(Xs,append_nat(cons_nat(X,nil_nat),Ys))),size_size_list_nat(Xs)) = X ).

% concat_inth
tff(fact_68_concat__inth,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] : aa_nat_int(nth_int(append_int(Xs,append_int(cons_int(X,nil_int),Ys))),size_size_list_int(Xs)) = X ).

% concat_inth
tff(fact_69_power__one,axiom,
    ! [N: nat] : power_power_rat(one_one_rat,N) = one_one_rat ).

% power_one
tff(fact_70_power__one,axiom,
    ! [N: nat] : aa_nat_nat(power_power_nat(one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_71_power__one,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),N) = one_one_real ).

% power_one
tff(fact_72_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_73_power__one,axiom,
    ! [N: nat] : power_power_complex(one_one_complex,N) = one_one_complex ).

% power_one
tff(fact_74_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(power_power_nat(A),one_one_nat) = A ).

% power_one_right
tff(fact_75_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_76_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_77_power__one__right,axiom,
    ! [A: complex] : power_power_complex(A,one_one_nat) = A ).

% power_one_right
tff(fact_78_nth__repl,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( ( M != N )
         => ( nth_VEBT_VEBT(append_VEBT_VEBT(take_VEBT_VEBT(N,Xs),append_VEBT_VEBT(cons_VEBT_VEBT(X,nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(N),one_one_nat),Xs))),M) = nth_VEBT_VEBT(Xs,M) ) ) ) ) ).

% nth_repl
tff(fact_79_nth__repl,axiom,
    ! [M: nat,Xs: list_bool,N: nat,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => ( ( M != N )
         => ( pp(aa_nat_bool(nth_bool(append_bool(take_bool(N,Xs),append_bool(cons_bool(X,nil_bool),drop_bool(aa_nat_nat(plus_plus_nat(N),one_one_nat),Xs)))),M))
          <=> pp(aa_nat_bool(nth_bool(Xs),M)) ) ) ) ) ).

% nth_repl
tff(fact_80_nth__repl,axiom,
    ! [M: nat,Xs: list_nat,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => ( ( M != N )
         => ( aa_nat_nat(nth_nat(append_nat(take_nat(N,Xs),append_nat(cons_nat(X,nil_nat),drop_nat(aa_nat_nat(plus_plus_nat(N),one_one_nat),Xs)))),M) = aa_nat_nat(nth_nat(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_81_nth__repl,axiom,
    ! [M: nat,Xs: list_int,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => ( ( M != N )
         => ( aa_nat_int(nth_int(append_int(take_int(N,Xs),append_int(cons_int(X,nil_int),drop_int(aa_nat_nat(plus_plus_nat(N),one_one_nat),Xs)))),M) = aa_nat_int(nth_int(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_82_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_83_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = numeral_numeral_real(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_84_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_85_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = numeral_numeral_nat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_86_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_87_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_88_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real(N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_89_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_90_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat(N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_91_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_92_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(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_93_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ord_less_rat(one_one_rat,A)
     => ( ( power_power_rat(A,M) = power_power_rat(A,N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_94_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(power_power_nat(A),M) = aa_nat_nat(power_power_nat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_95_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_96_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_97_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(ord_max_real(numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(ord_max_real(numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(1)
tff(fact_98_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ord_less_eq_rat(numeral_numeral_rat(U),numeral_numeral_rat(V))
       => ( ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = numeral_numeral_rat(V) ) )
      & ( ~ ord_less_eq_rat(numeral_numeral_rat(U),numeral_numeral_rat(V))
       => ( ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = numeral_numeral_rat(U) ) ) ) ).

% max_number_of(1)
tff(fact_99_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(U)),numeral_numeral_nat(V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(U)),numeral_numeral_nat(V)) = numeral_numeral_nat(V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(U)),numeral_numeral_nat(V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(U)),numeral_numeral_nat(V)) = numeral_numeral_nat(U) ) ) ) ).

% max_number_of(1)
tff(fact_100_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( ord_max_int(aa_num_int(numeral_numeral_int,U),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( ord_max_int(aa_num_int(numeral_numeral_int,U),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_101_max__0__1_I5_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_102_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_real(ord_max_real(one_one_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).

% max_0_1(5)
tff(fact_103_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(X)) = numeral_numeral_rat(X) ).

% max_0_1(5)
tff(fact_104_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),numeral_numeral_nat(X)) = numeral_numeral_nat(X) ).

% max_0_1(5)
tff(fact_105_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_int(one_one_int,aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(5)
tff(fact_106_max__0__1_I6_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_107_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_real(ord_max_real(numeral_numeral_real(X)),one_one_real) = numeral_numeral_real(X) ).

% max_0_1(6)
tff(fact_108_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_rat(numeral_numeral_rat(X),one_one_rat) = numeral_numeral_rat(X) ).

% max_0_1(6)
tff(fact_109_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(X)),one_one_nat) = numeral_numeral_nat(X) ).

% max_0_1(6)
tff(fact_110_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_int(aa_num_int(numeral_numeral_int,X),one_one_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(6)
tff(fact_111_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : plus_plus_complex(numera6690914467698888265omplex(V),plus_plus_complex(numera6690914467698888265omplex(W),Z)) = plus_plus_complex(numera6690914467698888265omplex(plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_112_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(plus_plus_real(numeral_numeral_real(V)),aa_real_real(plus_plus_real(numeral_numeral_real(W)),Z)) = aa_real_real(plus_plus_real(numeral_numeral_real(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_113_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : plus_plus_rat(numeral_numeral_rat(V),plus_plus_rat(numeral_numeral_rat(W),Z)) = plus_plus_rat(numeral_numeral_rat(plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_114_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(plus_plus_nat(numeral_numeral_nat(V)),aa_nat_nat(plus_plus_nat(numeral_numeral_nat(W)),Z)) = aa_nat_nat(plus_plus_nat(numeral_numeral_nat(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_115_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : plus_plus_int(aa_num_int(numeral_numeral_int,V),plus_plus_int(aa_num_int(numeral_numeral_int,W),Z)) = plus_plus_int(aa_num_int(numeral_numeral_int,plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_116_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_plus_complex(numera6690914467698888265omplex(M),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_117_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(plus_plus_real(numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_118_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N)) = numeral_numeral_rat(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_119_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(plus_plus_nat(numeral_numeral_nat(M)),numeral_numeral_nat(N)) = numeral_numeral_nat(plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_120_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_plus_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,plus_plus_num(M,N)) ).

% numeral_plus_numeral
tff(fact_121_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_122_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ord_less_eq_rat(numeral_numeral_rat(M),numeral_numeral_rat(N))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_123_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_124_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(ord_less_eq_num(M),N)) ) ).

% numeral_le_iff
tff(fact_125_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
      ( member8440522571783428010at_nat(A,collec3392354462482085612at_nat(P))
    <=> pp(aa_Pro2232559814647311250t_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_126_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_127_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_128_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_129_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( member_nat(A,collect_nat(P))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_130_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_131_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_132_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_133_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_134_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_135_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_136_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_137_Collect__cong,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X3: complex] :
          ( pp(aa_complex_bool(P,X3))
        <=> pp(aa_complex_bool(Q,X3)) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_138_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_139_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_140_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_141_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_142_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_143_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ord_less_rat(numeral_numeral_rat(M),numeral_numeral_rat(N))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_144_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_145_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(ord_less_num(M),N)) ) ).

% numeral_less_iff
tff(fact_146__C1_C,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "1"
tff(fact_147__C4_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X2),na))
        & ! [Xa: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),na)))
           => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(X2,Xa)),na)) ) ) ) ).

% "4.IH"(1)
tff(fact_148_add__One__commute,axiom,
    ! [N: num] : plus_plus_num(one,N) = plus_plus_num(N,one) ).

% add_One_commute
tff(fact_149_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa_num_bool(ord_less_eq_num(X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_150_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% is_num_normalize(1)
tff(fact_151_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% is_num_normalize(1)
tff(fact_152_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : plus_plus_int(plus_plus_int(A,B),C) = plus_plus_int(A,plus_plus_int(B,C)) ).

% is_num_normalize(1)
tff(fact_153_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_154_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat(one_one_rat,one_one_rat) ).

% le_numeral_extra(4)
tff(fact_155_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_156_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_157_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_158_less__numeral__extra_I4_J,axiom,
    ~ ord_less_rat(one_one_rat,one_one_rat) ).

% less_numeral_extra(4)
tff(fact_159_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_160_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_161_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),numeral_numeral_real(N))) ).

% one_le_numeral
tff(fact_162_one__le__numeral,axiom,
    ! [N: num] : ord_less_eq_rat(one_one_rat,numeral_numeral_rat(N)) ).

% one_le_numeral
tff(fact_163_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),numeral_numeral_nat(N))) ).

% one_le_numeral
tff(fact_164_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_165_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_166_not__numeral__less__one,axiom,
    ! [N: num] : ~ ord_less_rat(numeral_numeral_rat(N),one_one_rat) ).

% not_numeral_less_one
tff(fact_167_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_168_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_169_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = plus_plus_complex(numera6690914467698888265omplex(N),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_170_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit0,N)) = aa_real_real(plus_plus_real(numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_Bit0
tff(fact_171_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).

% numeral_Bit0
tff(fact_172_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_nat(aa_num_num(bit0,N)) = aa_nat_nat(plus_plus_nat(numeral_numeral_nat(N)),numeral_numeral_nat(N)) ).

% numeral_Bit0
tff(fact_173_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = plus_plus_int(aa_num_int(numeral_numeral_int,N),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_174_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_complex(one_one_complex,numera6690914467698888265omplex(X)) = plus_plus_complex(numera6690914467698888265omplex(X),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_175_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(plus_plus_real(one_one_real),numeral_numeral_real(X)) = aa_real_real(plus_plus_real(numeral_numeral_real(X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_176_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(X)) = plus_plus_rat(numeral_numeral_rat(X),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_177_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(plus_plus_nat(one_one_nat),numeral_numeral_nat(X)) = aa_nat_nat(plus_plus_nat(numeral_numeral_nat(X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_178_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_int(one_one_int,aa_num_int(numeral_numeral_int,X)) = plus_plus_int(aa_num_int(numeral_numeral_int,X),one_one_int) ).

% one_plus_numeral_commute
tff(fact_179_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_180_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_181_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_182_numeral__One,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numeral_One
tff(fact_183_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_184_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(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% one_le_power
tff(fact_185_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ord_less_eq_rat(one_one_rat,A)
     => ord_less_eq_rat(one_one_rat,power_power_rat(A,N)) ) ).

% one_le_power
tff(fact_186_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(power_power_nat(A),N))) ) ).

% one_le_power
tff(fact_187_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_188_numerals_I1_J,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numerals(1)
tff(fact_189_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
     => ( 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(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2))) ) ) ).

% power_strict_increasing
tff(fact_190_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
     => ( ord_less_rat(one_one_rat,A)
       => ord_less_rat(power_power_rat(A,N),power_power_rat(A,N2)) ) ) ).

% power_strict_increasing
tff(fact_191_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
     => ( 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(power_power_nat(A),N)),aa_nat_nat(power_power_nat(A),N2))) ) ) ).

% power_strict_increasing
tff(fact_192_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
     => ( 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),N2))) ) ) ).

% power_strict_increasing
tff(fact_193_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(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_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_194_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ord_less_rat(one_one_rat,A)
     => ( ord_less_rat(power_power_rat(A,M),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_195_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(power_power_nat(A),M)),aa_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_196_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_197_power__increasing,axiom,
    ! [N: nat,N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
     => ( 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(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2))) ) ) ).

% power_increasing
tff(fact_198_power__increasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
     => ( ord_less_eq_rat(one_one_rat,A)
       => ord_less_eq_rat(power_power_rat(A,N),power_power_rat(A,N2)) ) ) ).

% power_increasing
tff(fact_199_power__increasing,axiom,
    ! [N: nat,N2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
     => ( 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(power_power_nat(A),N)),aa_nat_nat(power_power_nat(A),N2))) ) ) ).

% power_increasing
tff(fact_200_power__increasing,axiom,
    ! [N: nat,N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
     => ( 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),N2))) ) ) ).

% power_increasing
tff(fact_201_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(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_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_202_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ord_less_rat(one_one_rat,A)
     => ( ord_less_eq_rat(power_power_rat(A,M),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_203_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(power_power_nat(A),M)),aa_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_204_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_205_one__power2,axiom,
    power_power_rat(one_one_rat,numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_206_one__power2,axiom,
    aa_nat_nat(power_power_nat(one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_207_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_208_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_209_one__power2,axiom,
    power_power_complex(one_one_complex,numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_210_less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% less_exp
tff(fact_211_nat__1__add__1,axiom,
    aa_nat_nat(plus_plus_nat(one_one_nat),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_212_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(power_power_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_213_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(power_power_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(power_power_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_214_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(power_power_nat(K),M))) ) ).

% self_le_ge2_pow
tff(fact_215_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K))
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(power_power_nat(B),N3)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(power_power_nat(B),aa_nat_nat(plus_plus_nat(N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_216_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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))
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(power_power_nat(B),N3)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(power_power_nat(B),aa_nat_nat(plus_plus_nat(N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_217__C11_C,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(append_VEBT_VEBT(take_VEBT_VEBT(aa_nat_nat(vEBT_VEBT_high(xa),na),treeList),append_VEBT_VEBT(cons_VEBT_VEBT(vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(vEBT_VEBT_low(xa),na)),nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(aa_nat_nat(vEBT_VEBT_high(xa),na)),one_one_nat),treeList))))))
     => pp(aa_nat_bool(vEBT_invar_vebt(X2),na)) ) ).

% "11"
tff(fact_218_nth__drop,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(drop_VEBT_VEBT(N,Xs),I2) = nth_VEBT_VEBT(Xs,aa_nat_nat(plus_plus_nat(N),I2)) ) ) ).

% nth_drop
tff(fact_219_nth__drop,axiom,
    ! [N: nat,Xs: list_bool,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(drop_bool(N,Xs)),I2))
      <=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(plus_plus_nat(N),I2))) ) ) ).

% nth_drop
tff(fact_220_nth__drop,axiom,
    ! [N: nat,Xs: list_nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(drop_nat(N,Xs)),I2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(plus_plus_nat(N),I2)) ) ) ).

% nth_drop
tff(fact_221_nth__drop,axiom,
    ! [N: nat,Xs: list_int,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(drop_int(N,Xs)),I2) = aa_nat_int(nth_int(Xs),aa_nat_nat(plus_plus_nat(N),I2)) ) ) ).

% nth_drop
tff(fact_222_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( nil_VEBT_VEBT = drop_VEBT_VEBT(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_223_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( nil_bool = drop_bool(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_224_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat = drop_nat(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_225_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( nil_int = drop_int(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_226_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( drop_VEBT_VEBT(N,Xs) = nil_VEBT_VEBT )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_227_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( drop_bool(N,Xs) = nil_bool )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_228_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( drop_nat(N,Xs) = nil_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_229_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( drop_int(N,Xs) = nil_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_230_drop__all,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),N))
     => ( drop_VEBT_VEBT(N,Xs) = nil_VEBT_VEBT ) ) ).

% drop_all
tff(fact_231_drop__all,axiom,
    ! [Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),N))
     => ( drop_bool(N,Xs) = nil_bool ) ) ).

% drop_all
tff(fact_232_drop__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),N))
     => ( drop_nat(N,Xs) = nil_nat ) ) ).

% drop_all
tff(fact_233_drop__all,axiom,
    ! [Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),N))
     => ( drop_int(N,Xs) = nil_int ) ) ).

% drop_all
tff(fact_234_nth__append__length__plus,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,N: nat] : nth_VEBT_VEBT(append_VEBT_VEBT(Xs,Ys),aa_nat_nat(plus_plus_nat(size_s6755466524823107622T_VEBT(Xs)),N)) = nth_VEBT_VEBT(Ys,N) ).

% nth_append_length_plus
tff(fact_235_nth__append__length__plus,axiom,
    ! [Xs: list_bool,Ys: list_bool,N: nat] :
      ( pp(aa_nat_bool(nth_bool(append_bool(Xs,Ys)),aa_nat_nat(plus_plus_nat(size_size_list_bool(Xs)),N)))
    <=> pp(aa_nat_bool(nth_bool(Ys),N)) ) ).

% nth_append_length_plus
tff(fact_236_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] : aa_nat_nat(nth_nat(append_nat(Xs,Ys)),aa_nat_nat(plus_plus_nat(size_size_list_nat(Xs)),N)) = aa_nat_nat(nth_nat(Ys),N) ).

% nth_append_length_plus
tff(fact_237_nth__append__length__plus,axiom,
    ! [Xs: list_int,Ys: list_int,N: nat] : aa_nat_int(nth_int(append_int(Xs,Ys)),aa_nat_nat(plus_plus_nat(size_size_list_int(Xs)),N)) = aa_nat_int(nth_int(Ys),N) ).

% nth_append_length_plus
tff(fact_238_nth__append__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT,Ys: list_VEBT_VEBT] : nth_VEBT_VEBT(append_VEBT_VEBT(Xs,cons_VEBT_VEBT(X,Ys)),size_s6755466524823107622T_VEBT(Xs)) = X ).

% nth_append_length
tff(fact_239_nth__append__length,axiom,
    ! [Xs: list_bool,X: bool,Ys: list_bool] :
      ( pp(aa_nat_bool(nth_bool(append_bool(Xs,cons_bool(X,Ys))),size_size_list_bool(Xs)))
    <=> pp(X) ) ).

% nth_append_length
tff(fact_240_nth__append__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] : aa_nat_nat(nth_nat(append_nat(Xs,cons_nat(X,Ys))),size_size_list_nat(Xs)) = X ).

% nth_append_length
tff(fact_241_nth__append__length,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] : aa_nat_int(nth_int(append_int(Xs,cons_int(X,Ys))),size_size_list_int(Xs)) = X ).

% nth_append_length
tff(fact_242_append__take__drop__id,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : append_VEBT_VEBT(take_VEBT_VEBT(N,Xs),drop_VEBT_VEBT(N,Xs)) = Xs ).

% append_take_drop_id
tff(fact_243_append__take__drop__id,axiom,
    ! [N: nat,Xs: list_int] : append_int(take_int(N,Xs),drop_int(N,Xs)) = Xs ).

% append_take_drop_id
tff(fact_244_append__take__drop__id,axiom,
    ! [N: nat,Xs: list_nat] : append_nat(take_nat(N,Xs),drop_nat(N,Xs)) = Xs ).

% append_take_drop_id
tff(fact_245_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(ord_less_num(one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_246_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_247__C12_C,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na))),vEBT_vebt_insert(summary,aa_nat_nat(vEBT_VEBT_high(xa),na)),summary)),na)) ).

% "12"
tff(fact_248_nth__take,axiom,
    ! [I2: nat,N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( aa_nat_int(nth_int(take_int(N,Xs)),I2) = aa_nat_int(nth_int(Xs),I2) ) ) ).

% nth_take
tff(fact_249_nth__take,axiom,
    ! [I2: nat,N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( nth_VEBT_VEBT(take_VEBT_VEBT(N,Xs),I2) = nth_VEBT_VEBT(Xs,I2) ) ) ).

% nth_take
tff(fact_250_nth__take,axiom,
    ! [I2: nat,N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( aa_nat_nat(nth_nat(take_nat(N,Xs)),I2) = aa_nat_nat(nth_nat(Xs),I2) ) ) ).

% nth_take
tff(fact_251_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_252_list_Oinject,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT,Y21: vEBT_VEBT,Y22: list_VEBT_VEBT] :
      ( ( cons_VEBT_VEBT(X21,X22) = cons_VEBT_VEBT(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_253_list_Oinject,axiom,
    ! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
      ( ( cons_int(X21,X22) = cons_int(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_254_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( cons_nat(X21,X22) = cons_nat(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_255_append_Oassoc,axiom,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,C: list_VEBT_VEBT] : append_VEBT_VEBT(append_VEBT_VEBT(A,B),C) = append_VEBT_VEBT(A,append_VEBT_VEBT(B,C)) ).

% append.assoc
tff(fact_256_append_Oassoc,axiom,
    ! [A: list_int,B: list_int,C: list_int] : append_int(append_int(A,B),C) = append_int(A,append_int(B,C)) ).

% append.assoc
tff(fact_257_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] : append_nat(append_nat(A,B),C) = append_nat(A,append_nat(B,C)) ).

% append.assoc
tff(fact_258_append__assoc,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_VEBT_VEBT] : append_VEBT_VEBT(append_VEBT_VEBT(Xs,Ys),Zs) = append_VEBT_VEBT(Xs,append_VEBT_VEBT(Ys,Zs)) ).

% append_assoc
tff(fact_259_append__assoc,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int] : append_int(append_int(Xs,Ys),Zs) = append_int(Xs,append_int(Ys,Zs)) ).

% append_assoc
tff(fact_260_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] : append_nat(append_nat(Xs,Ys),Zs) = append_nat(Xs,append_nat(Ys,Zs)) ).

% append_assoc
tff(fact_261_append__same__eq,axiom,
    ! [Ys: list_VEBT_VEBT,Xs: list_VEBT_VEBT,Zs: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Ys,Xs) = append_VEBT_VEBT(Zs,Xs) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_262_append__same__eq,axiom,
    ! [Ys: list_int,Xs: list_int,Zs: list_int] :
      ( ( append_int(Ys,Xs) = append_int(Zs,Xs) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_263_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( append_nat(Ys,Xs) = append_nat(Zs,Xs) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_264_same__append__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,Ys) = append_VEBT_VEBT(Xs,Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_265_same__append__eq,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ( append_int(Xs,Ys) = append_int(Xs,Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_266_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat(Xs,Ys) = append_nat(Xs,Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_267__C0_C,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => pp(aa_nat_bool(vEBT_invar_vebt(X2),na)) ) ).

% "0"
tff(fact_268_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_269_semiring__norm_I83_J,axiom,
    ! [N: num] : one != aa_num_num(bit0,N) ).

% semiring_norm(83)
tff(fact_270_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_271_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y2)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_272_append_Oright__neutral,axiom,
    ! [A: list_VEBT_VEBT] : append_VEBT_VEBT(A,nil_VEBT_VEBT) = A ).

% append.right_neutral
tff(fact_273_append_Oright__neutral,axiom,
    ! [A: list_int] : append_int(A,nil_int) = A ).

% append.right_neutral
tff(fact_274_append_Oright__neutral,axiom,
    ! [A: list_nat] : append_nat(A,nil_nat) = A ).

% append.right_neutral
tff(fact_275_append__Nil2,axiom,
    ! [Xs: list_VEBT_VEBT] : append_VEBT_VEBT(Xs,nil_VEBT_VEBT) = Xs ).

% append_Nil2
tff(fact_276_append__Nil2,axiom,
    ! [Xs: list_int] : append_int(Xs,nil_int) = Xs ).

% append_Nil2
tff(fact_277_append__Nil2,axiom,
    ! [Xs: list_nat] : append_nat(Xs,nil_nat) = Xs ).

% append_Nil2
tff(fact_278_append__self__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,Ys) = Xs )
    <=> ( Ys = nil_VEBT_VEBT ) ) ).

% append_self_conv
tff(fact_279_append__self__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( append_int(Xs,Ys) = Xs )
    <=> ( Ys = nil_int ) ) ).

% append_self_conv
tff(fact_280_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( append_nat(Xs,Ys) = Xs )
    <=> ( Ys = nil_nat ) ) ).

% append_self_conv
tff(fact_281_self__append__conv,axiom,
    ! [Y2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Y2 = append_VEBT_VEBT(Y2,Ys) )
    <=> ( Ys = nil_VEBT_VEBT ) ) ).

% self_append_conv
tff(fact_282_self__append__conv,axiom,
    ! [Y2: list_int,Ys: list_int] :
      ( ( Y2 = append_int(Y2,Ys) )
    <=> ( Ys = nil_int ) ) ).

% self_append_conv
tff(fact_283_self__append__conv,axiom,
    ! [Y2: list_nat,Ys: list_nat] :
      ( ( Y2 = append_nat(Y2,Ys) )
    <=> ( Ys = nil_nat ) ) ).

% self_append_conv
tff(fact_284_append__self__conv2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,Ys) = Ys )
    <=> ( Xs = nil_VEBT_VEBT ) ) ).

% append_self_conv2
tff(fact_285_append__self__conv2,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( append_int(Xs,Ys) = Ys )
    <=> ( Xs = nil_int ) ) ).

% append_self_conv2
tff(fact_286_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( append_nat(Xs,Ys) = Ys )
    <=> ( Xs = nil_nat ) ) ).

% append_self_conv2
tff(fact_287_self__append__conv2,axiom,
    ! [Y2: list_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( Y2 = append_VEBT_VEBT(Xs,Y2) )
    <=> ( Xs = nil_VEBT_VEBT ) ) ).

% self_append_conv2
tff(fact_288_self__append__conv2,axiom,
    ! [Y2: list_int,Xs: list_int] :
      ( ( Y2 = append_int(Xs,Y2) )
    <=> ( Xs = nil_int ) ) ).

% self_append_conv2
tff(fact_289_self__append__conv2,axiom,
    ! [Y2: list_nat,Xs: list_nat] :
      ( ( Y2 = append_nat(Xs,Y2) )
    <=> ( Xs = nil_nat ) ) ).

% self_append_conv2
tff(fact_290_Nil__is__append__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( nil_VEBT_VEBT = append_VEBT_VEBT(Xs,Ys) )
    <=> ( ( Xs = nil_VEBT_VEBT )
        & ( Ys = nil_VEBT_VEBT ) ) ) ).

% Nil_is_append_conv
tff(fact_291_Nil__is__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_int = append_int(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% Nil_is_append_conv
tff(fact_292_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat = append_nat(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
tff(fact_293_append__is__Nil__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,Ys) = nil_VEBT_VEBT )
    <=> ( ( Xs = nil_VEBT_VEBT )
        & ( Ys = nil_VEBT_VEBT ) ) ) ).

% append_is_Nil_conv
tff(fact_294_append__is__Nil__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( append_int(Xs,Ys) = nil_int )
    <=> ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% append_is_Nil_conv
tff(fact_295_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( append_nat(Xs,Ys) = nil_nat )
    <=> ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
tff(fact_296_append__eq__append__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Us: list_VEBT_VEBT,Vs: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        | ( size_s6755466524823107622T_VEBT(Us) = size_s6755466524823107622T_VEBT(Vs) ) )
     => ( ( append_VEBT_VEBT(Xs,Us) = append_VEBT_VEBT(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_297_append__eq__append__conv,axiom,
    ! [Xs: list_bool,Ys: list_bool,Us: list_bool,Vs: list_bool] :
      ( ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        | ( size_size_list_bool(Us) = size_size_list_bool(Vs) ) )
     => ( ( append_bool(Xs,Us) = append_bool(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_298_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        | ( size_size_list_nat(Us) = size_size_list_nat(Vs) ) )
     => ( ( append_nat(Xs,Us) = append_nat(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_299_append__eq__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
      ( ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        | ( size_size_list_int(Us) = size_size_list_int(Vs) ) )
     => ( ( append_int(Xs,Us) = append_int(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_300_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : plus_plus_num(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_num_num(bit0,plus_plus_num(M,N)) ).

% semiring_norm(6)
tff(fact_301_drop__drop,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBT] : drop_VEBT_VEBT(N,drop_VEBT_VEBT(M,Xs)) = drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(N),M),Xs) ).

% drop_drop
tff(fact_302_drop__drop,axiom,
    ! [N: nat,M: nat,Xs: list_nat] : drop_nat(N,drop_nat(M,Xs)) = drop_nat(aa_nat_nat(plus_plus_nat(N),M),Xs) ).

% drop_drop
tff(fact_303_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(71)
tff(fact_304_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(78)
tff(fact_305_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa_num_bool(ord_less_eq_num(one),N)) ).

% semiring_norm(68)
tff(fact_306_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(ord_less_num(M),one)) ).

% semiring_norm(75)
tff(fact_307_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(power_power_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_308__C4_OIH_C_I2_J,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m)))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(summary,X)),m)) ) ).

% "4.IH"(2)
tff(fact_309_myIHs,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X),na))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),na)))
         => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(X,Xa2)),na)) ) ) ) ).

% myIHs
tff(fact_310_append1__eq__conv,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,cons_VEBT_VEBT(X,nil_VEBT_VEBT)) = append_VEBT_VEBT(Ys,cons_VEBT_VEBT(Y2,nil_VEBT_VEBT)) )
    <=> ( ( Xs = Ys )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
tff(fact_311_append1__eq__conv,axiom,
    ! [Xs: list_int,X: int,Ys: list_int,Y2: int] :
      ( ( append_int(Xs,cons_int(X,nil_int)) = append_int(Ys,cons_int(Y2,nil_int)) )
    <=> ( ( Xs = Ys )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
tff(fact_312_append1__eq__conv,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y2: nat] :
      ( ( append_nat(Xs,cons_nat(X,nil_nat)) = append_nat(Ys,cons_nat(Y2,nil_nat)) )
    <=> ( ( Xs = Ys )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
tff(fact_313_length__append,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(append_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(plus_plus_nat(size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_append
tff(fact_314_length__append,axiom,
    ! [Xs: list_bool,Ys: list_bool] : size_size_list_bool(append_bool(Xs,Ys)) = aa_nat_nat(plus_plus_nat(size_size_list_bool(Xs)),size_size_list_bool(Ys)) ).

% length_append
tff(fact_315_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_size_list_nat(append_nat(Xs,Ys)) = aa_nat_nat(plus_plus_nat(size_size_list_nat(Xs)),size_size_list_nat(Ys)) ).

% length_append
tff(fact_316_length__append,axiom,
    ! [Xs: list_int,Ys: list_int] : size_size_list_int(append_int(Xs,Ys)) = aa_nat_nat(plus_plus_nat(size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_append
tff(fact_317_semiring__norm_I2_J,axiom,
    plus_plus_num(one,one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_318_take__all,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),N))
     => ( take_VEBT_VEBT(N,Xs) = Xs ) ) ).

% take_all
tff(fact_319_take__all,axiom,
    ! [Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),N))
     => ( take_bool(N,Xs) = Xs ) ) ).

% take_all
tff(fact_320_take__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),N))
     => ( take_nat(N,Xs) = Xs ) ) ).

% take_all
tff(fact_321_take__all,axiom,
    ! [Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),N))
     => ( take_int(N,Xs) = Xs ) ) ).

% take_all
tff(fact_322_take__all__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( take_VEBT_VEBT(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),N)) ) ).

% take_all_iff
tff(fact_323_take__all__iff,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( take_bool(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),N)) ) ).

% take_all_iff
tff(fact_324_take__all__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( take_nat(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),N)) ) ).

% take_all_iff
tff(fact_325_take__all__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( take_int(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),N)) ) ).

% take_all_iff
tff(fact_326_not__Cons__self2,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : cons_VEBT_VEBT(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_327_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] : cons_int(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_328_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] : cons_nat(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_329_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),B2)
    <=> ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,set_Pr5648618587558075414at_nat(Xs))
         => member8440522571783428010at_nat(X4,B2) ) ) ).

% subset_code(1)
tff(fact_330_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B2)
    <=> ! [X4: complex] :
          ( member_complex(X4,set_complex2(Xs))
         => member_complex(X4,B2) ) ) ).

% subset_code(1)
tff(fact_331_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B2)
    <=> ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => member_real(X4,B2) ) ) ).

% subset_code(1)
tff(fact_332_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),B2)) ) ) ).

% subset_code(1)
tff(fact_333_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(set_nat2(Xs),B2)
    <=> ! [X4: nat] :
          ( member_nat(X4,set_nat2(Xs))
         => member_nat(X4,B2) ) ) ).

% subset_code(1)
tff(fact_334_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B2)
    <=> ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => member_int(X4,B2) ) ) ).

% subset_code(1)
tff(fact_335_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_336_neq__if__length__neq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) != size_size_list_bool(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_337_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_338_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_339_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).

% Ex_list_of_length
tff(fact_340_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_341_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_342_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_343_append__eq__appendI,axiom,
    ! [Xs: list_VEBT_VEBT,Xs1: list_VEBT_VEBT,Zs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Us: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,Xs1) = Zs )
     => ( ( Ys = append_VEBT_VEBT(Xs1,Us) )
       => ( append_VEBT_VEBT(Xs,Ys) = append_VEBT_VEBT(Zs,Us) ) ) ) ).

% append_eq_appendI
tff(fact_344_append__eq__appendI,axiom,
    ! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us: list_int] :
      ( ( append_int(Xs,Xs1) = Zs )
     => ( ( Ys = append_int(Xs1,Us) )
       => ( append_int(Xs,Ys) = append_int(Zs,Us) ) ) ) ).

% append_eq_appendI
tff(fact_345_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( append_nat(Xs,Xs1) = Zs )
     => ( ( Ys = append_nat(Xs1,Us) )
       => ( append_nat(Xs,Ys) = append_nat(Zs,Us) ) ) ) ).

% append_eq_appendI
tff(fact_346_append__eq__append__conv2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_VEBT_VEBT,Ts: list_VEBT_VEBT] :
      ( ( append_VEBT_VEBT(Xs,Ys) = append_VEBT_VEBT(Zs,Ts) )
    <=> ? [Us2: list_VEBT_VEBT] :
          ( ( ( Xs = append_VEBT_VEBT(Zs,Us2) )
            & ( append_VEBT_VEBT(Us2,Ys) = Ts ) )
          | ( ( append_VEBT_VEBT(Xs,Us2) = Zs )
            & ( Ys = append_VEBT_VEBT(Us2,Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_347_append__eq__append__conv2,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
      ( ( append_int(Xs,Ys) = append_int(Zs,Ts) )
    <=> ? [Us2: list_int] :
          ( ( ( Xs = append_int(Zs,Us2) )
            & ( append_int(Us2,Ys) = Ts ) )
          | ( ( append_int(Xs,Us2) = Zs )
            & ( Ys = append_int(Us2,Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_348_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( append_nat(Xs,Ys) = append_nat(Zs,Ts) )
    <=> ? [Us2: list_nat] :
          ( ( ( Xs = append_nat(Zs,Us2) )
            & ( append_nat(Us2,Ys) = Ts ) )
          | ( ( append_nat(Xs,Us2) = Zs )
            & ( Ys = append_nat(Us2,Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_349_take__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ! [I3: nat] : take_VEBT_VEBT(I3,Xs) = take_VEBT_VEBT(I3,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_350_take__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ! [I3: nat] : take_nat(I3,Xs) = take_nat(I3,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_351_list_Odistinct_I1_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] : nil_VEBT_VEBT != cons_VEBT_VEBT(X21,X22) ).

% list.distinct(1)
tff(fact_352_list_Odistinct_I1_J,axiom,
    ! [X21: int,X22: list_int] : nil_int != cons_int(X21,X22) ).

% list.distinct(1)
tff(fact_353_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : nil_nat != cons_nat(X21,X22) ).

% list.distinct(1)
tff(fact_354_list_OdiscI,axiom,
    ! [List: list_VEBT_VEBT,X21: vEBT_VEBT,X22: list_VEBT_VEBT] :
      ( ( List = cons_VEBT_VEBT(X21,X22) )
     => ( List != nil_VEBT_VEBT ) ) ).

% list.discI
tff(fact_355_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X22: list_int] :
      ( ( List = cons_int(X21,X22) )
     => ( List != nil_int ) ) ).

% list.discI
tff(fact_356_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List = cons_nat(X21,X22) )
     => ( List != nil_nat ) ) ).

% list.discI
tff(fact_357_list_Oexhaust,axiom,
    ! [Y2: list_VEBT_VEBT] :
      ( ( Y2 != nil_VEBT_VEBT )
     => ~ ! [X212: vEBT_VEBT,X222: list_VEBT_VEBT] : Y2 != cons_VEBT_VEBT(X212,X222) ) ).

% list.exhaust
tff(fact_358_list_Oexhaust,axiom,
    ! [Y2: list_int] :
      ( ( Y2 != nil_int )
     => ~ ! [X212: int,X222: list_int] : Y2 != cons_int(X212,X222) ) ).

% list.exhaust
tff(fact_359_list_Oexhaust,axiom,
    ! [Y2: list_nat] :
      ( ( Y2 != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] : Y2 != cons_nat(X212,X222) ) ).

% list.exhaust
tff(fact_360_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X3: int,Xs2: list_int] : X != cons_int(X3,Xs2)
     => ( X = nil_int ) ) ).

% min_list.cases
tff(fact_361_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat,Xs2: list_nat] : X != cons_nat(X3,Xs2)
     => ( X = nil_nat ) ) ).

% min_list.cases
tff(fact_362_transpose_Ocases,axiom,
    ! [X: list_list_VEBT_VEBT] :
      ( ( X != nil_list_VEBT_VEBT )
     => ( ! [Xss: list_list_VEBT_VEBT] : X != cons_list_VEBT_VEBT(nil_VEBT_VEBT,Xss)
       => ~ ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Xss: list_list_VEBT_VEBT] : X != cons_list_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_363_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] : X != cons_list_int(nil_int,Xss)
       => ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] : X != cons_list_int(cons_int(X3,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_364_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] : X != cons_list_nat(nil_nat,Xss)
       => ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] : X != cons_list_nat(cons_nat(X3,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_365_remdups__adj_Ocases,axiom,
    ! [X: list_VEBT_VEBT] :
      ( ( X != nil_VEBT_VEBT )
     => ( ! [X3: vEBT_VEBT] : X != cons_VEBT_VEBT(X3,nil_VEBT_VEBT)
       => ~ ! [X3: vEBT_VEBT,Y3: vEBT_VEBT,Xs2: list_VEBT_VEBT] : X != cons_VEBT_VEBT(X3,cons_VEBT_VEBT(Y3,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_366_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X3: int] : X != cons_int(X3,nil_int)
       => ~ ! [X3: int,Y3: int,Xs2: list_int] : X != cons_int(X3,cons_int(Y3,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_367_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X3: nat] : X != cons_nat(X3,nil_nat)
       => ~ ! [X3: nat,Y3: nat,Xs2: list_nat] : X != cons_nat(X3,cons_nat(Y3,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_368_neq__Nil__conv,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( Xs != nil_VEBT_VEBT )
    <=> ? [Y4: vEBT_VEBT,Ys2: list_VEBT_VEBT] : Xs = cons_VEBT_VEBT(Y4,Ys2) ) ).

% neq_Nil_conv
tff(fact_369_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
    <=> ? [Y4: int,Ys2: list_int] : Xs = cons_int(Y4,Ys2) ) ).

% neq_Nil_conv
tff(fact_370_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
    <=> ? [Y4: nat,Ys2: list_nat] : Xs = cons_nat(Y4,Ys2) ) ).

% neq_Nil_conv
tff(fact_371_list__induct2_H,axiom,
    ! [P: fun_li5407204298491981421T_bool,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,nil_VEBT_VEBT),nil_VEBT_VEBT))
     => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] : pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,cons_VEBT_VEBT(X3,Xs2)),nil_VEBT_VEBT))
       => ( ! [Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] : pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,nil_VEBT_VEBT),cons_VEBT_VEBT(Y3,Ys3)))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
                ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,Xs2),Ys3))
               => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3))) )
           => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_372_list__induct2_H,axiom,
    ! [P: fun_li1500669845994789693t_bool,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,nil_VEBT_VEBT),nil_int))
     => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] : pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,cons_VEBT_VEBT(X3,Xs2)),nil_int))
       => ( ! [Y3: int,Ys3: list_int] : pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,nil_VEBT_VEBT),cons_int(Y3,Ys3)))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: int,Ys3: list_int] :
                ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,Xs2),Ys3))
               => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_int(Y3,Ys3))) )
           => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_373_list__induct2_H,axiom,
    ! [P: fun_li8419635973868945049t_bool,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,nil_VEBT_VEBT),nil_nat))
     => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] : pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,cons_VEBT_VEBT(X3,Xs2)),nil_nat))
       => ( ! [Y3: nat,Ys3: list_nat] : pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,nil_VEBT_VEBT),cons_nat(Y3,Ys3)))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: nat,Ys3: list_nat] :
                ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,Xs2),Ys3))
               => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_nat(Y3,Ys3))) )
           => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_374_list__induct2_H,axiom,
    ! [P: fun_li8993770084906592985T_bool,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(P,nil_int),nil_VEBT_VEBT))
     => ( ! [X3: int,Xs2: list_int] : pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(P,cons_int(X3,Xs2)),nil_VEBT_VEBT))
       => ( ! [Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] : pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(P,nil_int),cons_VEBT_VEBT(Y3,Ys3)))
         => ( ! [X3: int,Xs2: list_int,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
                ( pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(P,Xs2),Ys3))
               => pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(P,cons_int(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3))) )
           => pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_375_list__induct2_H,axiom,
    ! [P: fun_li582272206764200913t_bool,Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),nil_int))
     => ( ! [X3: int,Xs2: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X3,Xs2)),nil_int))
       => ( ! [Y3: int,Ys3: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),cons_int(Y3,Ys3)))
         => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
                ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs2),Ys3))
               => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X3,Xs2)),cons_int(Y3,Ys3))) )
           => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_376_list__induct2_H,axiom,
    ! [P: fun_li7501238334638356269t_bool,Xs: list_int,Ys: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),nil_nat))
     => ( ! [X3: int,Xs2: list_int] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X3,Xs2)),nil_nat))
       => ( ! [Y3: nat,Ys3: list_nat] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),cons_nat(Y3,Ys3)))
         => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat] :
                ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs2),Ys3))
               => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X3,Xs2)),cons_nat(Y3,Ys3))) )
           => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_377_list__induct2_H,axiom,
    ! [P: fun_li4519670091647362685T_bool,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,nil_nat),nil_VEBT_VEBT))
     => ( ! [X3: nat,Xs2: list_nat] : pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,cons_nat(X3,Xs2)),nil_VEBT_VEBT))
       => ( ! [Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] : pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,nil_nat),cons_VEBT_VEBT(Y3,Ys3)))
         => ( ! [X3: nat,Xs2: list_nat,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
                ( pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,Xs2),Ys3))
               => pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,cons_nat(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3))) )
           => pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_378_list__induct2_H,axiom,
    ! [P: fun_li1679355187408199469t_bool,Xs: list_nat,Ys: list_int] :
      ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),nil_int))
     => ( ! [X3: nat,Xs2: list_nat] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X3,Xs2)),nil_int))
       => ( ! [Y3: int,Ys3: list_int] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),cons_int(Y3,Ys3)))
         => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int] :
                ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs2),Ys3))
               => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3))) )
           => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_379_list__induct2_H,axiom,
    ! [P: fun_li8598321315282354825t_bool,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),nil_nat))
     => ( ! [X3: nat,Xs2: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X3,Xs2)),nil_nat))
       => ( ! [Y3: nat,Ys3: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),cons_nat(Y3,Ys3)))
         => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
                ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs2),Ys3))
               => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X3,Xs2)),cons_nat(Y3,Ys3))) )
           => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_380_list__nonempty__induct,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_li6006399982103069984T_bool] :
      ( ( Xs != nil_VEBT_VEBT )
     => ( ! [X3: vEBT_VEBT] : pp(aa_lis2822252094023462681T_bool(P,cons_VEBT_VEBT(X3,nil_VEBT_VEBT)))
       => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
              ( ( Xs2 != nil_VEBT_VEBT )
             => ( pp(aa_lis2822252094023462681T_bool(P,Xs2))
               => pp(aa_lis2822252094023462681T_bool(P,cons_VEBT_VEBT(X3,Xs2))) ) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_381_list__nonempty__induct,axiom,
    ! [Xs: list_int,P: fun_list_int_bool] :
      ( ( Xs != nil_int )
     => ( ! [X3: int] : pp(aa_list_int_bool(P,cons_int(X3,nil_int)))
       => ( ! [X3: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( pp(aa_list_int_bool(P,Xs2))
               => pp(aa_list_int_bool(P,cons_int(X3,Xs2))) ) )
         => pp(aa_list_int_bool(P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_382_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: fun_list_nat_bool] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : pp(aa_list_nat_bool(P,cons_nat(X3,nil_nat)))
       => ( ! [X3: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( pp(aa_list_nat_bool(P,Xs2))
               => pp(aa_list_nat_bool(P,cons_nat(X3,Xs2))) ) )
         => pp(aa_list_nat_bool(P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_383_list_Oset__intros_I2_J,axiom,
    ! [Y2: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( member8440522571783428010at_nat(Y2,set_Pr5648618587558075414at_nat(X22))
     => member8440522571783428010at_nat(Y2,set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X21,X22))) ) ).

% list.set_intros(2)
tff(fact_384_list_Oset__intros_I2_J,axiom,
    ! [Y2: complex,X22: list_complex,X21: complex] :
      ( member_complex(Y2,set_complex2(X22))
     => member_complex(Y2,set_complex2(cons_complex(X21,X22))) ) ).

% list.set_intros(2)
tff(fact_385_list_Oset__intros_I2_J,axiom,
    ! [Y2: real,X22: list_real,X21: real] :
      ( member_real(Y2,set_real2(X22))
     => member_real(Y2,set_real2(cons_real(X21,X22))) ) ).

% list.set_intros(2)
tff(fact_386_list_Oset__intros_I2_J,axiom,
    ! [Y2: vEBT_VEBT,X22: list_VEBT_VEBT,X21: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(X22)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(cons_VEBT_VEBT(X21,X22)))) ) ).

% list.set_intros(2)
tff(fact_387_list_Oset__intros_I2_J,axiom,
    ! [Y2: int,X22: list_int,X21: int] :
      ( member_int(Y2,set_int2(X22))
     => member_int(Y2,set_int2(cons_int(X21,X22))) ) ).

% list.set_intros(2)
tff(fact_388_list_Oset__intros_I2_J,axiom,
    ! [Y2: nat,X22: list_nat,X21: nat] :
      ( member_nat(Y2,set_nat2(X22))
     => member_nat(Y2,set_nat2(cons_nat(X21,X22))) ) ).

% list.set_intros(2)
tff(fact_389_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : member8440522571783428010at_nat(X21,set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X21,X22))) ).

% list.set_intros(1)
tff(fact_390_list_Oset__intros_I1_J,axiom,
    ! [X21: complex,X22: list_complex] : member_complex(X21,set_complex2(cons_complex(X21,X22))) ).

% list.set_intros(1)
tff(fact_391_list_Oset__intros_I1_J,axiom,
    ! [X21: real,X22: list_real] : member_real(X21,set_real2(cons_real(X21,X22))) ).

% list.set_intros(1)
tff(fact_392_list_Oset__intros_I1_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X21),set_VEBT_VEBT2(cons_VEBT_VEBT(X21,X22)))) ).

% list.set_intros(1)
tff(fact_393_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X22: list_int] : member_int(X21,set_int2(cons_int(X21,X22))) ).

% list.set_intros(1)
tff(fact_394_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : member_nat(X21,set_nat2(cons_nat(X21,X22))) ).

% list.set_intros(1)
tff(fact_395_list_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A: list_P6011104703257516679at_nat] :
      ( member8440522571783428010at_nat(E,set_Pr5648618587558075414at_nat(A))
     => ( ! [Z2: list_P6011104703257516679at_nat] : A != cons_P6512896166579812791at_nat(E,Z2)
       => ~ ! [Z1: product_prod_nat_nat,Z2: list_P6011104703257516679at_nat] :
              ( ( A = cons_P6512896166579812791at_nat(Z1,Z2) )
             => ~ member8440522571783428010at_nat(E,set_Pr5648618587558075414at_nat(Z2)) ) ) ) ).

% list.set_cases
tff(fact_396_list_Oset__cases,axiom,
    ! [E: complex,A: list_complex] :
      ( member_complex(E,set_complex2(A))
     => ( ! [Z2: list_complex] : A != cons_complex(E,Z2)
       => ~ ! [Z1: complex,Z2: list_complex] :
              ( ( A = cons_complex(Z1,Z2) )
             => ~ member_complex(E,set_complex2(Z2)) ) ) ) ).

% list.set_cases
tff(fact_397_list_Oset__cases,axiom,
    ! [E: real,A: list_real] :
      ( member_real(E,set_real2(A))
     => ( ! [Z2: list_real] : A != cons_real(E,Z2)
       => ~ ! [Z1: real,Z2: list_real] :
              ( ( A = cons_real(Z1,Z2) )
             => ~ member_real(E,set_real2(Z2)) ) ) ) ).

% list.set_cases
tff(fact_398_list_Oset__cases,axiom,
    ! [E: vEBT_VEBT,A: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,E),set_VEBT_VEBT2(A)))
     => ( ! [Z2: list_VEBT_VEBT] : A != cons_VEBT_VEBT(E,Z2)
       => ~ ! [Z1: vEBT_VEBT,Z2: list_VEBT_VEBT] :
              ( ( A = cons_VEBT_VEBT(Z1,Z2) )
             => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,E),set_VEBT_VEBT2(Z2))) ) ) ) ).

% list.set_cases
tff(fact_399_list_Oset__cases,axiom,
    ! [E: int,A: list_int] :
      ( member_int(E,set_int2(A))
     => ( ! [Z2: list_int] : A != cons_int(E,Z2)
       => ~ ! [Z1: int,Z2: list_int] :
              ( ( A = cons_int(Z1,Z2) )
             => ~ member_int(E,set_int2(Z2)) ) ) ) ).

% list.set_cases
tff(fact_400_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( member_nat(E,set_nat2(A))
     => ( ! [Z2: list_nat] : A != cons_nat(E,Z2)
       => ~ ! [Z1: nat,Z2: list_nat] :
              ( ( A = cons_nat(Z1,Z2) )
             => ~ member_nat(E,set_nat2(Z2)) ) ) ) ).

% list.set_cases
tff(fact_401_set__ConsD,axiom,
    ! [Y2: product_prod_nat_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8440522571783428010at_nat(Y2,set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X,Xs)))
     => ( ( Y2 = X )
        | member8440522571783428010at_nat(Y2,set_Pr5648618587558075414at_nat(Xs)) ) ) ).

% set_ConsD
tff(fact_402_set__ConsD,axiom,
    ! [Y2: complex,X: complex,Xs: list_complex] :
      ( member_complex(Y2,set_complex2(cons_complex(X,Xs)))
     => ( ( Y2 = X )
        | member_complex(Y2,set_complex2(Xs)) ) ) ).

% set_ConsD
tff(fact_403_set__ConsD,axiom,
    ! [Y2: real,X: real,Xs: list_real] :
      ( member_real(Y2,set_real2(cons_real(X,Xs)))
     => ( ( Y2 = X )
        | member_real(Y2,set_real2(Xs)) ) ) ).

% set_ConsD
tff(fact_404_set__ConsD,axiom,
    ! [Y2: vEBT_VEBT,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(cons_VEBT_VEBT(X,Xs))))
     => ( ( Y2 = X )
        | pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(Xs))) ) ) ).

% set_ConsD
tff(fact_405_set__ConsD,axiom,
    ! [Y2: int,X: int,Xs: list_int] :
      ( member_int(Y2,set_int2(cons_int(X,Xs)))
     => ( ( Y2 = X )
        | member_int(Y2,set_int2(Xs)) ) ) ).

% set_ConsD
tff(fact_406_set__ConsD,axiom,
    ! [Y2: nat,X: nat,Xs: list_nat] :
      ( member_nat(Y2,set_nat2(cons_nat(X,Xs)))
     => ( ( Y2 = X )
        | member_nat(Y2,set_nat2(Xs)) ) ) ).

% set_ConsD
tff(fact_407_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(cons_VEBT_VEBT(X,Xs))) ).

% set_subset_Cons
tff(fact_408_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ord_less_eq_set_nat(set_nat2(Xs),set_nat2(cons_nat(X,Xs))) ).

% set_subset_Cons
tff(fact_409_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : ord_less_eq_set_int(set_int2(Xs),set_int2(cons_int(X,Xs))) ).

% set_subset_Cons
tff(fact_410_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys4: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys4)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys4)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_411_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys4: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys4)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P,Ys4)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_412_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys4: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys4)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P,Ys4)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_413_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys4: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys4)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P,Ys4)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_414_append__Cons,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : append_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),Ys) = cons_VEBT_VEBT(X,append_VEBT_VEBT(Xs,Ys)) ).

% append_Cons
tff(fact_415_append__Cons,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] : append_int(cons_int(X,Xs),Ys) = cons_int(X,append_int(Xs,Ys)) ).

% append_Cons
tff(fact_416_append__Cons,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] : append_nat(cons_nat(X,Xs),Ys) = cons_nat(X,append_nat(Xs,Ys)) ).

% append_Cons
tff(fact_417_Cons__eq__appendI,axiom,
    ! [X: vEBT_VEBT,Xs1: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Xs: list_VEBT_VEBT,Zs: list_VEBT_VEBT] :
      ( ( cons_VEBT_VEBT(X,Xs1) = Ys )
     => ( ( Xs = append_VEBT_VEBT(Xs1,Zs) )
       => ( cons_VEBT_VEBT(X,Xs) = append_VEBT_VEBT(Ys,Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_418_Cons__eq__appendI,axiom,
    ! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
      ( ( cons_int(X,Xs1) = Ys )
     => ( ( Xs = append_int(Xs1,Zs) )
       => ( cons_int(X,Xs) = append_int(Ys,Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_419_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( cons_nat(X,Xs1) = Ys )
     => ( ( Xs = append_nat(Xs1,Zs) )
       => ( cons_nat(X,Xs) = append_nat(Ys,Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_420_append__Nil,axiom,
    ! [Ys: list_VEBT_VEBT] : append_VEBT_VEBT(nil_VEBT_VEBT,Ys) = Ys ).

% append_Nil
tff(fact_421_append__Nil,axiom,
    ! [Ys: list_int] : append_int(nil_int,Ys) = Ys ).

% append_Nil
tff(fact_422_append__Nil,axiom,
    ! [Ys: list_nat] : append_nat(nil_nat,Ys) = Ys ).

% append_Nil
tff(fact_423_append_Oleft__neutral,axiom,
    ! [A: list_VEBT_VEBT] : append_VEBT_VEBT(nil_VEBT_VEBT,A) = A ).

% append.left_neutral
tff(fact_424_append_Oleft__neutral,axiom,
    ! [A: list_int] : append_int(nil_int,A) = A ).

% append.left_neutral
tff(fact_425_append_Oleft__neutral,axiom,
    ! [A: list_nat] : append_nat(nil_nat,A) = A ).

% append.left_neutral
tff(fact_426_eq__Nil__appendI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
     => ( Xs = append_VEBT_VEBT(nil_VEBT_VEBT,Ys) ) ) ).

% eq_Nil_appendI
tff(fact_427_eq__Nil__appendI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
     => ( Xs = append_int(nil_int,Ys) ) ) ).

% eq_Nil_appendI
tff(fact_428_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs = append_nat(nil_nat,Ys) ) ) ).

% eq_Nil_appendI
tff(fact_429_take__Nil,axiom,
    ! [N: nat] : take_VEBT_VEBT(N,nil_VEBT_VEBT) = nil_VEBT_VEBT ).

% take_Nil
tff(fact_430_take__Nil,axiom,
    ! [N: nat] : take_int(N,nil_int) = nil_int ).

% take_Nil
tff(fact_431_take__Nil,axiom,
    ! [N: nat] : take_nat(N,nil_nat) = nil_nat ).

% take_Nil
tff(fact_432_in__set__takeD,axiom,
    ! [X: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
      ( member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(take_P2173866234530122223at_nat(N,Xs)))
     => member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(Xs)) ) ).

% in_set_takeD
tff(fact_433_in__set__takeD,axiom,
    ! [X: complex,N: nat,Xs: list_complex] :
      ( member_complex(X,set_complex2(take_complex(N,Xs)))
     => member_complex(X,set_complex2(Xs)) ) ).

% in_set_takeD
tff(fact_434_in__set__takeD,axiom,
    ! [X: real,N: nat,Xs: list_real] :
      ( member_real(X,set_real2(take_real(N,Xs)))
     => member_real(X,set_real2(Xs)) ) ).

% in_set_takeD
tff(fact_435_in__set__takeD,axiom,
    ! [X: vEBT_VEBT,N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(take_VEBT_VEBT(N,Xs))))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs))) ) ).

% in_set_takeD
tff(fact_436_in__set__takeD,axiom,
    ! [X: int,N: nat,Xs: list_int] :
      ( member_int(X,set_int2(take_int(N,Xs)))
     => member_int(X,set_int2(Xs)) ) ).

% in_set_takeD
tff(fact_437_in__set__takeD,axiom,
    ! [X: nat,N: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(take_nat(N,Xs)))
     => member_nat(X,set_nat2(Xs)) ) ).

% in_set_takeD
tff(fact_438_set__take__subset,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(take_VEBT_VEBT(N,Xs)),set_VEBT_VEBT2(Xs)) ).

% set_take_subset
tff(fact_439_set__take__subset,axiom,
    ! [N: nat,Xs: list_nat] : ord_less_eq_set_nat(set_nat2(take_nat(N,Xs)),set_nat2(Xs)) ).

% set_take_subset
tff(fact_440_set__take__subset,axiom,
    ! [N: nat,Xs: list_int] : ord_less_eq_set_int(set_int2(take_int(N,Xs)),set_int2(Xs)) ).

% set_take_subset
tff(fact_441_drop__Nil,axiom,
    ! [N: nat] : drop_VEBT_VEBT(N,nil_VEBT_VEBT) = nil_VEBT_VEBT ).

% drop_Nil
tff(fact_442_drop__Nil,axiom,
    ! [N: nat] : drop_int(N,nil_int) = nil_int ).

% drop_Nil
tff(fact_443_drop__Nil,axiom,
    ! [N: nat] : drop_nat(N,nil_nat) = nil_nat ).

% drop_Nil
tff(fact_444_in__set__dropD,axiom,
    ! [X: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
      ( member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(drop_P8868858903918902087at_nat(N,Xs)))
     => member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(Xs)) ) ).

% in_set_dropD
tff(fact_445_in__set__dropD,axiom,
    ! [X: complex,N: nat,Xs: list_complex] :
      ( member_complex(X,set_complex2(drop_complex(N,Xs)))
     => member_complex(X,set_complex2(Xs)) ) ).

% in_set_dropD
tff(fact_446_in__set__dropD,axiom,
    ! [X: real,N: nat,Xs: list_real] :
      ( member_real(X,set_real2(drop_real(N,Xs)))
     => member_real(X,set_real2(Xs)) ) ).

% in_set_dropD
tff(fact_447_in__set__dropD,axiom,
    ! [X: vEBT_VEBT,N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(drop_VEBT_VEBT(N,Xs))))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs))) ) ).

% in_set_dropD
tff(fact_448_in__set__dropD,axiom,
    ! [X: int,N: nat,Xs: list_int] :
      ( member_int(X,set_int2(drop_int(N,Xs)))
     => member_int(X,set_int2(Xs)) ) ).

% in_set_dropD
tff(fact_449_in__set__dropD,axiom,
    ! [X: nat,N: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(drop_nat(N,Xs)))
     => member_nat(X,set_nat2(Xs)) ) ).

% in_set_dropD
tff(fact_450_set__drop__subset,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(drop_VEBT_VEBT(N,Xs)),set_VEBT_VEBT2(Xs)) ).

% set_drop_subset
tff(fact_451_set__drop__subset,axiom,
    ! [N: nat,Xs: list_nat] : ord_less_eq_set_nat(set_nat2(drop_nat(N,Xs)),set_nat2(Xs)) ).

% set_drop_subset
tff(fact_452_set__drop__subset,axiom,
    ! [N: nat,Xs: list_int] : ord_less_eq_set_int(set_int2(drop_int(N,Xs)),set_int2(Xs)) ).

% set_drop_subset
tff(fact_453_list__induct2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: fun_li5407204298491981421T_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,nil_VEBT_VEBT),nil_VEBT_VEBT))
       => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
              ( ( size_s6755466524823107622T_VEBT(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
             => ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,Xs2),Ys3))
               => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3))) ) )
         => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_454_list__induct2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,P: fun_li8202109546025615434l_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(P,nil_VEBT_VEBT),nil_bool))
       => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: bool,Ys3: list_bool] :
              ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_bool(Ys3) )
             => ( pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(P,Xs2),Ys3))
               => pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_bool(Y3,Ys3))) ) )
         => pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_455_list__induct2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,P: fun_li8419635973868945049t_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,nil_VEBT_VEBT),nil_nat))
       => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: nat,Ys3: list_nat] :
              ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_nat(Ys3) )
             => ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,Xs2),Ys3))
               => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_nat(Y3,Ys3))) ) )
         => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_456_list__induct2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,P: fun_li1500669845994789693t_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,nil_VEBT_VEBT),nil_int))
       => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: int,Ys3: list_int] :
              ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_int(Ys3) )
             => ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,Xs2),Ys3))
               => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_int(Y3,Ys3))) ) )
         => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_457_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,P: fun_li149329434408825296T_bool] :
      ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(P,nil_bool),nil_VEBT_VEBT))
       => ( ! [X3: bool,Xs2: list_bool,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
              ( ( size_size_list_bool(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
             => ( pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(P,Xs2),Ys3))
               => pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(P,cons_bool(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3))) ) )
         => pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_458_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_bool,P: fun_li402716648095076269l_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,nil_bool),nil_bool))
       => ( ! [X3: bool,Xs2: list_bool,Y3: bool,Ys3: list_bool] :
              ( ( size_size_list_bool(Xs2) = size_size_list_bool(Ys3) )
             => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,Xs2),Ys3))
               => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,cons_bool(X3,Xs2)),cons_bool(Y3,Ys3))) ) )
         => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_459_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_nat,P: fun_li1802912181200660086t_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,nil_bool),nil_nat))
       => ( ! [X3: bool,Xs2: list_bool,Y3: nat,Ys3: list_nat] :
              ( ( size_size_list_bool(Xs2) = size_size_list_nat(Ys3) )
             => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,Xs2),Ys3))
               => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,cons_bool(X3,Xs2)),cons_nat(Y3,Ys3))) ) )
         => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_460_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_int,P: fun_li4107318090181280538t_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,nil_bool),nil_int))
       => ( ! [X3: bool,Xs2: list_bool,Y3: int,Ys3: list_int] :
              ( ( size_size_list_bool(Xs2) = size_size_list_int(Ys3) )
             => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,Xs2),Ys3))
               => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,cons_bool(X3,Xs2)),cons_int(Y3,Ys3))) ) )
         => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_461_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT,P: fun_li4519670091647362685T_bool] :
      ( ( size_size_list_nat(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,nil_nat),nil_VEBT_VEBT))
       => ( ! [X3: nat,Xs2: list_nat,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
              ( ( size_size_list_nat(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
             => ( pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,Xs2),Ys3))
               => pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,cons_nat(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3))) ) )
         => pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_462_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_bool,P: fun_li6875962081873459290l_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,nil_nat),nil_bool))
       => ( ! [X3: nat,Xs2: list_nat,Y3: bool,Ys3: list_bool] :
              ( ( size_size_list_nat(Xs2) = size_size_list_bool(Ys3) )
             => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,Xs2),Ys3))
               => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,cons_nat(X3,Xs2)),cons_bool(Y3,Ys3))) ) )
         => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_463_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_VEBT_VEBT,P: fun_li6608604213202121590T_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ( size_s6755466524823107622T_VEBT(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis1171169218965586543T_bool(P,nil_VEBT_VEBT),nil_VEBT_VEBT),nil_VEBT_VEBT))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
               => ( ( size_s6755466524823107622T_VEBT(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                 => ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis1171169218965586543T_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis1171169218965586543T_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2))) ) ) )
           => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis1171169218965586543T_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_464_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_bool,P: fun_li4259300718648188627l_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ( size_s6755466524823107622T_VEBT(Ys) = size_size_list_bool(Zs) )
       => ( pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis7824369692577881164l_bool(P,nil_VEBT_VEBT),nil_VEBT_VEBT),nil_bool))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT,Z3: bool,Zs2: list_bool] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
               => ( ( size_s6755466524823107622T_VEBT(Ys3) = size_size_list_bool(Zs2) )
                 => ( pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis7824369692577881164l_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis7824369692577881164l_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3)),cons_bool(Z3,Zs2))) ) ) )
           => pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis7824369692577881164l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_465_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_nat,P: fun_li7006304056279447142t_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ( size_s6755466524823107622T_VEBT(Ys) = size_size_list_nat(Zs) )
       => ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis7873390175868453399t_bool(P,nil_VEBT_VEBT),nil_VEBT_VEBT),nil_nat))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT,Z3: nat,Zs2: list_nat] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
               => ( ( size_s6755466524823107622T_VEBT(Ys3) = size_size_list_nat(Zs2) )
                 => ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis7873390175868453399t_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis7873390175868453399t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3)),cons_nat(Z3,Zs2))) ) ) )
           => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis7873390175868453399t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_466_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Zs: list_int,P: fun_li7692880089319997706t_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ( size_s6755466524823107622T_VEBT(Ys) = size_size_list_int(Zs) )
       => ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis954424047994298043t_bool(P,nil_VEBT_VEBT),nil_VEBT_VEBT),nil_int))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: vEBT_VEBT,Ys3: list_VEBT_VEBT,Z3: int,Zs2: list_int] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_s6755466524823107622T_VEBT(Ys3) )
               => ( ( size_s6755466524823107622T_VEBT(Ys3) = size_size_list_int(Zs2) )
                 => ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis954424047994298043t_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis954424047994298043t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_VEBT_VEBT(Y3,Ys3)),cons_int(Z3,Zs2))) ) ) )
           => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis954424047994298043t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_467_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,Zs: list_VEBT_VEBT,P: fun_li472734531553275225T_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( ( size_size_list_bool(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis8994961617815866834T_bool(P,nil_VEBT_VEBT),nil_bool),nil_VEBT_VEBT))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: bool,Ys3: list_bool,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_bool(Ys3) )
               => ( ( size_size_list_bool(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                 => ( pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis8994961617815866834T_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis8994961617815866834T_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_bool(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2))) ) ) )
           => pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis8994961617815866834T_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_468_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,Zs: list_bool,P: fun_li5971970583006782646l_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( ( size_size_list_bool(Ys) = size_size_list_bool(Zs) )
       => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis5684437626337506223l_bool(P,nil_VEBT_VEBT),nil_bool),nil_bool))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: bool,Ys3: list_bool,Z3: bool,Zs2: list_bool] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_bool(Ys3) )
               => ( ( size_size_list_bool(Ys3) = size_size_list_bool(Zs2) )
                 => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis5684437626337506223l_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis5684437626337506223l_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_bool(Y3,Ys3)),cons_bool(Z3,Zs2))) ) ) )
           => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis5684437626337506223l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_469_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,Zs: list_nat,P: fun_li6473728550467749827t_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( ( size_size_list_bool(Ys) = size_size_list_nat(Zs) )
       => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis7253160690866659828t_bool(P,nil_VEBT_VEBT),nil_bool),nil_nat))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: bool,Ys3: list_bool,Z3: nat,Zs2: list_nat] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_bool(Ys3) )
               => ( ( size_size_list_bool(Ys3) = size_size_list_nat(Zs2) )
                 => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis7253160690866659828t_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis7253160690866659828t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_bool(Y3,Ys3)),cons_nat(Z3,Zs2))) ) ) )
           => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis7253160690866659828t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_470_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,Zs: list_int,P: fun_li7160304583508300391t_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( ( size_size_list_bool(Ys) = size_size_list_int(Zs) )
       => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis334194562992504472t_bool(P,nil_VEBT_VEBT),nil_bool),nil_int))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: bool,Ys3: list_bool,Z3: int,Zs2: list_int] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_bool(Ys3) )
               => ( ( size_size_list_bool(Ys3) = size_size_list_int(Zs2) )
                 => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis334194562992504472t_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis334194562992504472t_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_bool(Y3,Ys3)),cons_int(Z3,Zs2))) ) ) )
           => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis334194562992504472t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_471_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Zs: list_VEBT_VEBT,P: fun_li8272695713475243082T_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(aa_lis3973424293646871035T_bool(P,nil_VEBT_VEBT),nil_nat),nil_VEBT_VEBT))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: nat,Ys3: list_nat,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_nat(Ys3) )
               => ( ( size_size_list_nat(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                 => ( pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(aa_lis3973424293646871035T_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(aa_lis3973424293646871035T_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_nat(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2))) ) ) )
           => pp(aa_lis2822252094023462681T_bool(aa_lis1494331557287675838T_bool(aa_lis3973424293646871035T_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_472_list__induct3,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Zs: list_bool,P: fun_li6373356552795678631l_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_bool(Zs) )
       => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis3102838554684683224l_bool(P,nil_VEBT_VEBT),nil_nat),nil_bool))
         => ( ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y3: nat,Ys3: list_nat,Z3: bool,Zs2: list_bool] :
                ( ( size_s6755466524823107622T_VEBT(Xs2) = size_size_list_nat(Ys3) )
               => ( ( size_size_list_nat(Ys3) = size_size_list_bool(Zs2) )
                 => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis3102838554684683224l_bool(P,Xs2),Ys3),Zs2))
                   => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis3102838554684683224l_bool(P,cons_VEBT_VEBT(X3,Xs2)),cons_nat(Y3,Ys3)),cons_bool(Z3,Zs2))) ) ) )
           => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis3102838554684683224l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_473_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_VEBT_VEBT,P: fun_li3210304982093000467T_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_s6755466524823107622T_VEBT(Ws) )
         => ( pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(aa_lis1609274531354153591T_bool(aa_lis5302737185788338004T_bool(P,nil_nat),nil_nat),nil_int),nil_VEBT_VEBT))
           => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat,Z3: int,Zs2: list_int,W2: vEBT_VEBT,Ws2: list_VEBT_VEBT] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_s6755466524823107622T_VEBT(Ws2) )
                     => ( pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(aa_lis1609274531354153591T_bool(aa_lis5302737185788338004T_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(aa_lis1609274531354153591T_bool(aa_lis5302737185788338004T_bool(P,cons_nat(X3,Xs2)),cons_nat(Y3,Ys3)),cons_int(Z3,Zs2)),cons_VEBT_VEBT(W2,Ws2))) ) ) ) )
             => pp(aa_lis2822252094023462681T_bool(aa_lis5968431550546906138T_bool(aa_lis1609274531354153591T_bool(aa_lis5302737185788338004T_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_474_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_bool,P: fun_li50010890099246064l_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_bool(Ws) )
         => ( pp(aa_list_bool_bool(aa_lis8375160404872846455l_bool(aa_lis5688191147790683220l_bool(aa_lis8164832519907246769l_bool(P,nil_nat),nil_nat),nil_int),nil_bool))
           => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat,Z3: int,Zs2: list_int,W2: bool,Ws2: list_bool] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_bool(Ws2) )
                     => ( pp(aa_list_bool_bool(aa_lis8375160404872846455l_bool(aa_lis5688191147790683220l_bool(aa_lis8164832519907246769l_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_bool_bool(aa_lis8375160404872846455l_bool(aa_lis5688191147790683220l_bool(aa_lis8164832519907246769l_bool(P,cons_nat(X3,Xs2)),cons_nat(Y3,Ys3)),cons_int(Z3,Zs2)),cons_bool(W2,Ws2))) ) ) ) )
             => pp(aa_list_bool_bool(aa_lis8375160404872846455l_bool(aa_lis5688191147790683220l_bool(aa_lis8164832519907246769l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_475_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_nat,P: fun_li4602333441959630751t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_nat(Ws) )
         => ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,nil_nat),nil_nat),nil_int),nil_nat))
           => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat,Z3: int,Zs2: list_int,W2: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_nat(Ws2) )
                     => ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,cons_nat(X3,Xs2)),cons_nat(Y3,Ys3)),cons_int(Z3,Zs2)),cons_nat(W2,Ws2))) ) ) ) )
             => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_476_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_int,P: fun_li6998123803554500675t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_int(Ws) )
         => ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,nil_nat),nil_nat),nil_int),nil_int))
           => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat,Z3: int,Zs2: list_int,W2: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_int(Ws2) )
                     => ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,cons_nat(X3,Xs2)),cons_nat(Y3,Ys3)),cons_int(Z3,Zs2)),cons_int(W2,Ws2))) ) ) ) )
             => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_477_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_VEBT_VEBT,Ws: list_VEBT_VEBT,P: fun_li3583542845167048579T_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( ( size_s6755466524823107622T_VEBT(Zs) = size_s6755466524823107622T_VEBT(Ws) )
         => ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis2365136247091392243T_bool(aa_lis6296824781172769516T_bool(P,nil_nat),nil_int),nil_VEBT_VEBT),nil_VEBT_VEBT))
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT,W2: vEBT_VEBT,Ws2: list_VEBT_VEBT] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                   => ( ( size_s6755466524823107622T_VEBT(Zs2) = size_s6755466524823107622T_VEBT(Ws2) )
                     => ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis2365136247091392243T_bool(aa_lis6296824781172769516T_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis2365136247091392243T_bool(aa_lis6296824781172769516T_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2)),cons_VEBT_VEBT(W2,Ws2))) ) ) ) )
             => pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aa_lis2365136247091392243T_bool(aa_lis6296824781172769516T_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_478_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_VEBT_VEBT,Ws: list_bool,P: fun_li3847990264654146656l_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( ( size_s6755466524823107622T_VEBT(Zs) = size_size_list_bool(Ws) )
         => ( pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis9015153175320828112l_bool(aa_lis8586777610711615049l_bool(P,nil_nat),nil_int),nil_VEBT_VEBT),nil_bool))
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT,W2: bool,Ws2: list_bool] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                   => ( ( size_s6755466524823107622T_VEBT(Zs2) = size_size_list_bool(Ws2) )
                     => ( pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis9015153175320828112l_bool(aa_lis8586777610711615049l_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis9015153175320828112l_bool(aa_lis8586777610711615049l_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2)),cons_bool(W2,Ws2))) ) ) ) )
             => pp(aa_list_bool_bool(aa_lis8849893572050924539l_bool(aa_lis9015153175320828112l_bool(aa_lis8586777610711615049l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_479_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_VEBT_VEBT,Ws: list_nat,P: fun_li3572619897263067951t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( ( size_s6755466524823107622T_VEBT(Zs) = size_size_list_nat(Ws) )
         => ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis3860336716327511187t_bool(aa_lis5452854915921875312t_bool(P,nil_nat),nil_int),nil_VEBT_VEBT),nil_nat))
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT,W2: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                   => ( ( size_s6755466524823107622T_VEBT(Zs2) = size_size_list_nat(Ws2) )
                     => ( pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis3860336716327511187t_bool(aa_lis5452854915921875312t_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis3860336716327511187t_bool(aa_lis5452854915921875312t_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2)),cons_nat(W2,Ws2))) ) ) ) )
             => pp(aa_list_nat_bool(aa_lis8884807844152417042t_bool(aa_lis3860336716327511187t_bool(aa_lis5452854915921875312t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_480_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_VEBT_VEBT,Ws: list_int,P: fun_li5968410258857937875t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_s6755466524823107622T_VEBT(Zs) )
       => ( ( size_s6755466524823107622T_VEBT(Zs) = size_size_list_int(Ws) )
         => ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis6164742625308131639t_bool(aa_lis6139430948962425876t_bool(P,nil_nat),nil_int),nil_VEBT_VEBT),nil_int))
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z3: vEBT_VEBT,Zs2: list_VEBT_VEBT,W2: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_s6755466524823107622T_VEBT(Zs2) )
                   => ( ( size_s6755466524823107622T_VEBT(Zs2) = size_size_list_int(Ws2) )
                     => ( pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis6164742625308131639t_bool(aa_lis6139430948962425876t_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis6164742625308131639t_bool(aa_lis6139430948962425876t_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3)),cons_VEBT_VEBT(Z3,Zs2)),cons_int(W2,Ws2))) ) ) ) )
             => pp(aa_list_int_bool(aa_lis8424524892505770422t_bool(aa_lis6164742625308131639t_bool(aa_lis6139430948962425876t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_481_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_bool,Ws: list_VEBT_VEBT,P: fun_li2591207965931870182T_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_bool(Zs) )
       => ( ( size_size_list_bool(Zs) = size_s6755466524823107622T_VEBT(Ws) )
         => ( pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis962373063704037974T_bool(aa_lis4800211423616701647T_bool(P,nil_nat),nil_int),nil_bool),nil_VEBT_VEBT))
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z3: bool,Zs2: list_bool,W2: vEBT_VEBT,Ws2: list_VEBT_VEBT] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_bool(Zs2) )
                   => ( ( size_size_list_bool(Zs2) = size_s6755466524823107622T_VEBT(Ws2) )
                     => ( pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis962373063704037974T_bool(aa_lis4800211423616701647T_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis962373063704037974T_bool(aa_lis4800211423616701647T_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3)),cons_bool(Z3,Zs2)),cons_VEBT_VEBT(W2,Ws2))) ) ) ) )
             => pp(aa_lis2822252094023462681T_bool(aa_lis3566239262678054657T_bool(aa_lis962373063704037974T_bool(aa_lis4800211423616701647T_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_482_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_bool,Ws: list_bool,P: fun_li6530564641721891011l_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_bool(Zs) )
       => ( ( size_size_list_bool(Zs) = size_size_list_bool(Ws) )
         => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis2259609000592408627l_bool(aa_lis7827536655497258028l_bool(P,nil_nat),nil_int),nil_bool),nil_bool))
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z3: bool,Zs2: list_bool,W2: bool,Ws2: list_bool] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_bool(Zs2) )
                   => ( ( size_size_list_bool(Zs2) = size_size_list_bool(Ws2) )
                     => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis2259609000592408627l_bool(aa_lis7827536655497258028l_bool(P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis2259609000592408627l_bool(aa_lis7827536655497258028l_bool(P,cons_nat(X3,Xs2)),cons_int(Y3,Ys3)),cons_bool(Z3,Zs2)),cons_bool(W2,Ws2))) ) ) ) )
             => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis2259609000592408627l_bool(aa_lis7827536655497258028l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_483_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y2))
           => ( pp(aa_nat_bool(vEBT_vebt_member(T),Y2))
              | ( X = Y2 ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_484_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ) ) ).

% member_bound
tff(fact_485_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(vEBT_VEBT_bit_concat(H,L),D) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_486_low__inv,axiom,
    ! [X: nat,N: nat,Y2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(vEBT_VEBT_low(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_487_high__inv,axiom,
    ! [X: nat,N: nat,Y2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))),X)),N) = Y2 ) ) ).

% high_inv
tff(fact_488__C9_C,axiom,
    aa_nat_nat(divide_divide_nat(deg),numeral_numeral_nat(aa_num_num(bit0,one))) = na ).

% "9"
tff(fact_489_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(plus_plus_nat(K),M)),aa_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_490_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] : X != aa_nat_nat(plus_plus_nat(N3),N3)
     => ~ ! [N3: nat] : X != aa_nat_nat(plus_plus_nat(N3),aa_nat_nat(suc,N3)) ) ).

% even_odd_cases
tff(fact_491_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_492_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_493_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_494_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y23) )
    <=> ( X23 = Y23 ) ) ).

% nat.inject
tff(fact_495_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_496_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> member_nat(X,vEBT_set_vebt(T)) ) ) ).

% member_correct
tff(fact_497_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(plus_plus_nat(A),B))),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),A)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_498_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(vEBT_VEBT_high(X),N) = aa_nat_nat(divide_divide_nat(X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_499_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_500_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_501_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_502_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_503_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(plus_plus_nat(M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M),N)) ).

% add_Suc_right
tff(fact_504_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(plus_plus_nat(K),M)),aa_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_505_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_506_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_507_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_508_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(plus_plus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_509_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,numeral_numeral_nat(N)) = numeral_numeral_nat(plus_plus_num(N,one)) ).

% Suc_numeral
tff(fact_510_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(plus_plus_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_511_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(plus_plus_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_512_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_513_Suc__inject,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y2) )
     => ( X = Y2 ) ) ).

% Suc_inject
tff(fact_514_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_515_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_516_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_517_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_518_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(plus_plus_nat(N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_519_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_520_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => pp(aa_nat_bool(P,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I3)))
               => pp(aa_nat_bool(P,I3)) ) )
         => pp(aa_nat_bool(P,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_521_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),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,I3),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J)) ) ) ) ).

% less_Suc_induct
tff(fact_522_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),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,I2)),K)) ) ) ).

% less_trans_Suc
tff(fact_523_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_524_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_525_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))
    <=> ? [M2: nat] :
          ( ( M = aa_nat_nat(suc,M2) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% Suc_less_eq2
tff(fact_526_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_527_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_528_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_529_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_530_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_531_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_532_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_533_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_534_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_535_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
     => ( ( K != aa_nat_nat(suc,I2) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_536_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,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X3),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X3),Z3)) ) )
         => ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N3),aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_537_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))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N3))
             => ( pp(aa_nat_bool(P,N3))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_538_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M3)),N3))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N3)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_539_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_540_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_541_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_542_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M4))
     => ? [M5: nat] : M4 = aa_nat_nat(suc,M5) ) ).

% Suc_le_D
tff(fact_543_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_544_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_545_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_546_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),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),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_547_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_548_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_549_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_550_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_551_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(suc,M)),N) = aa_nat_nat(plus_plus_nat(M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_552_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M),N)) ).

% add_Suc
tff(fact_553_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(plus_plus_nat(K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(plus_plus_nat(K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_554_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(plus_plus_nat(M),N)) = aa_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_555_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(M),N)),K) = aa_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_556_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_557_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_558_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = 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),Q2)) ).

% nat_mult_max_right
tff(fact_559_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_560_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(I2),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_561_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_562_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_563_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_564_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_565_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_566_inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N3)))
                 => pp(aa_nat_bool(P,N3)) ) ) )
         => pp(aa_nat_bool(P,I2)) ) ) ) ).

% inc_induct
tff(fact_567_dec__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(P,I2))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J))
               => ( pp(aa_nat_bool(P,N3))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_568_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_569_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_570_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(plus_plus_nat(M),K2)) ) ).

% less_imp_Suc_add
tff(fact_571_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(plus_plus_nat(M),K3)) ) ).

% less_iff_Suc_add
tff(fact_572_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M),I2)))) ).

% less_add_Suc2
tff(fact_573_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(I2),M)))) ).

% less_add_Suc1
tff(fact_574_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M),Q3)) ) ).

% less_natE
tff(fact_575_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(plus_plus_nat(one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_576_plus__1__eq__Suc,axiom,
    plus_plus_nat(one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_577_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(plus_plus_nat(N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_578_linorder__neqE__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y2),X)) ) ) ).

% linorder_neqE_nat
tff(fact_579_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ~ pp(aa_nat_bool(P,N3))
         => ? [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
              & ~ pp(aa_nat_bool(P,M3)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_580_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N3)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_581_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_582_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_583_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_584_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_585_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_586_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))
            & ! [Y: nat] :
                ( pp(aa_nat_bool(P,Y))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_587_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_588_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_589_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_590_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),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,I2),K)) ) ) ).

% le_trans
tff(fact_591_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_592_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(numeral_numeral_nat(V)),N)) = aa_nat_nat(plus_plus_nat(numeral_numeral_nat(plus_plus_num(V,one))),N) ).

% Suc_nat_number_of_add
tff(fact_593_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_594_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_595_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_596_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_597_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_598_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_599_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(plus_plus_nat(M),L) = aa_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_600_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(plus_plus_nat(M),J))) ) ).

% trans_less_add2
tff(fact_601_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(plus_plus_nat(J),M))) ) ).

% trans_less_add1
tff(fact_602_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K)),aa_nat_nat(plus_plus_nat(J),K))) ) ).

% add_less_mono1
tff(fact_603_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(J),I2)),I2)) ).

% not_add_less2
tff(fact_604_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),J)),I2)) ).

% not_add_less1
tff(fact_605_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),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(plus_plus_nat(I2),K)),aa_nat_nat(plus_plus_nat(J),L))) ) ) ).

% add_less_mono
tff(fact_606_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K)) ) ).

% add_lessD1
tff(fact_607_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(plus_plus_nat(M),K3) ) ).

% nat_le_iff_add
tff(fact_608_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(plus_plus_nat(M),J))) ) ).

% trans_le_add2
tff(fact_609_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(plus_plus_nat(J),M))) ) ).

% trans_le_add1
tff(fact_610_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(I2),K)),aa_nat_nat(plus_plus_nat(J),K))) ) ).

% add_le_mono1
tff(fact_611_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),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(plus_plus_nat(I2),K)),aa_nat_nat(plus_plus_nat(J),L))) ) ) ).

% add_le_mono
tff(fact_612_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N3: nat] : L = aa_nat_nat(plus_plus_nat(K),N3) ) ).

% le_Suc_ex
tff(fact_613_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(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_614_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(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_615_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(plus_plus_nat(M),N))) ).

% le_add2
tff(fact_616_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(plus_plus_nat(N),M))) ).

% le_add1
tff(fact_617_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(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_618_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(plus_plus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(plus_plus_nat(M),N)),aa_nat_nat(plus_plus_nat(M),Q2)) ).

% nat_add_max_right
tff(fact_619_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(plus_plus_nat(M),Q2)),aa_nat_nat(plus_plus_nat(N),Q2)) ).

% nat_add_max_left
tff(fact_620_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M5: nat,N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M5)),aa_nat_nat(F,N3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(plus_plus_nat(M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_621_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(plus_plus_nat(M),M)),numeral_numeral_nat(aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_622_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,M))),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_623_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_624_div__nat__eqI,axiom,
    ! [N: nat,Q2: 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),Q2)),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,Q2))))
       => ( aa_nat_nat(divide_divide_nat(M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_625_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N))) ) ).

% enat_ord_number(1)
tff(fact_626_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N))) ) ).

% enat_ord_number(2)
tff(fact_627_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_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,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).

% double_not_eq_Suc_double
tff(fact_628_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_629_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))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_630_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : times_times_num(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,times_times_num(M,N))) ).

% semiring_norm(13)
tff(fact_631_semiring__norm_I12_J,axiom,
    ! [N: num] : times_times_num(one,N) = N ).

% semiring_norm(12)
tff(fact_632_semiring__norm_I11_J,axiom,
    ! [M: num] : times_times_num(M,one) = M ).

% semiring_norm(11)
tff(fact_633_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_634_num__double,axiom,
    ! [N: num] : times_times_num(aa_num_num(bit0,one),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_635_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M3: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M3),N3))
             => pp(aa_Ext2772979874139974381t_bool(P,M3)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N3)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).

% enat_less_induct
tff(fact_636_four__x__squared,axiom,
    ! [X: real] : aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),X)),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_637_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(times_times_real(A),C))),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),numeral_numeral_nat(aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_638_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_639_div__le__mono,axiom,
    ! [M: nat,N: nat,K: 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(divide_divide_nat(M),K)),aa_nat_nat(divide_divide_nat(N),K))) ) ).

% div_le_mono
tff(fact_640_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(M),N)),M)) ).

% div_le_dividend
tff(fact_641_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(divide_divide_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(divide_divide_nat(M),N)),Q2) ).

% div_mult2_eq
tff(fact_642_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(divide_divide_nat(M),N)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_643_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: 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,I2),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(divide_divide_nat(M),N)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_644_div__times__less__eq__dividend,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(times_times_nat,aa_nat_nat(divide_divide_nat(M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_645_times__div__less__eq__dividend,axiom,
    ! [N: 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),aa_nat_nat(divide_divide_nat(M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_646_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,numeral_numeral_nat(aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) ).

% Suc_double_not_eq_double
tff(fact_647_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),A)),aa_real_real(times_times_real(R2),R2)) = aa_real_real(divide_divide_real(A),R2) ).

% real_divide_square_eq
tff(fact_648_less__eq__real__def,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2))
        | ( X = Y2 ) ) ) ).

% less_eq_real_def
tff(fact_649_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X2: real] : member_real(X2,S3)
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( member_real(X3,S3)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z4)) )
       => ? [Y3: real] :
            ( ! [X2: real] :
                ( member_real(X2,S3)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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_650_real__arch__pow,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N3))) ) ).

% real_arch_pow
tff(fact_651_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(vEBT_VEBT_low(X),N) = aa_nat_nat(modulo_modulo_nat(X),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_652_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( 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(TreeList,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(Deg),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(Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_653_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_654_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: 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(Mi),Ma)),Deg,TreeList,Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Deg))
        & ( ( X = Mi )
          | ( 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,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),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(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ) ) ).

% member_inv
tff(fact_655__C10_C,axiom,
    vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xa),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(vEBT_VEBT_low(xa),na))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na))),vEBT_vebt_insert(summary,aa_nat_nat(vEBT_VEBT_high(xa),na)),summary)) ).

% "10"
tff(fact_656_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_657_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_658_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_659_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = divide_divide_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_660_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_661_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_662_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_663_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_664_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_665_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(plus_plus_nat(M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_666_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(plus_plus_nat(M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_678_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) )
          & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_679_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_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_680_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_681_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_682_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_683_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_684_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(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_685_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_686_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_687_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_688_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_689_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_690_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(times_times_real(X),X)))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_691_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(divide_divide_nat(M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_692_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_693_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(divide_divide_nat(M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_694_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_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_695_power__Suc__0,axiom,
    ! [N: nat] : aa_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_696_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(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_697_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_698_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_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(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(divide_divide_nat(M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_699_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(modulo_modulo_nat(M),N) = M ) ) ).

% mod_less
tff(fact_700_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: 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(Mi),Ma)),Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_701_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_702_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_703_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_704_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(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_705_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(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_706_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: 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(Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_707_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_708_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_709_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_710_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_711_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Mi),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(Mi),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(Mi),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(vEBT_VEBT_low(Mi),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(Mi),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(vEBT_VEBT_high(Mi),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_712_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_713_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_714_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,M))),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_715_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(plus_plus_nat(M),M)),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_716_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( numeral_numeral_nat(K) != one_one_nat )
     => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K)),N))),numeral_numeral_nat(K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_717_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_718_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_719_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(modulo_modulo_nat(M),N)) = N )
       => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(modulo_modulo_nat(M),N)) != N )
       => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(modulo_modulo_nat(M),N)) ) ) ) ).

% mod_Suc
tff(fact_720_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(modulo_modulo_nat(M),N)),N)) ) ).

% mod_less_divisor
tff(fact_721_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_722_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(modulo_modulo_nat(M),N)),N)) ) ).

% mod_le_divisor
tff(fact_723_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))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_724_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(aa_real_fun_nat_real(power_power_real,X3),N) = A )
            & ! [Y: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = A ) )
               => ( Y = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_725_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(modulo_modulo_nat(M),N)))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_726_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(modulo_modulo_nat(M),N))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_727_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(modulo_modulo_nat(M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_728_num_Osize_I4_J,axiom,
    size_size_num(one) = zero_zero_nat ).

% num.size(4)
tff(fact_729_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% not0_implies_Suc
tff(fact_730_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_731_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_732_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_733_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P,N3)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_734_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_735_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,N3))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_736_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y2 != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_737_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa_nat_nat(suc,X23) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_738_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_739_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_740_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).

% nat.distinct(1)
tff(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( ~ pp(aa_nat_bool(P,N3))
             => ? [M3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
                  & ~ pp(aa_nat_bool(P,M3)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_748_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_749_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_750_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_751_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_752_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(plus_plus_nat(zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_753_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(plus_plus_nat(M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_754_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_755_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_756_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y2: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(X),N) = aa_nat_nat(modulo_modulo_nat(Y2),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(plus_plus_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(plus_plus_nat(Y2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_757_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))
     => ( divide_divide_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),B)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),A)) = divide_divide_int(plus_plus_int(B,one_one_int),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_758_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))
     => ( divide_divide_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),B)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),A)) = divide_divide_int(B,A) ) ) ).

% pos_zdiv_mult_2
tff(fact_759_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,N)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_760_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_761_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(modulo_modulo_nat(M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_762_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),zero_z5237406670263579293d_enat))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_763_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N)) ).

% i0_lb
tff(fact_764_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(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_765_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P2: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P2))
         => ( ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),P2))
               => ( pp(aa_nat_bool(P,N3))
                 => pp(aa_nat_bool(P,aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,N3)),P2))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_766_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(modulo_modulo_nat(M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_767_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),Q2) = aa_nat_nat(modulo_modulo_nat(N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S2: nat] : M != aa_nat_nat(plus_plus_nat(N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_768_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),Q2) = aa_nat_nat(modulo_modulo_nat(N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S2: nat] : N != aa_nat_nat(plus_plus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_769_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y2: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(X),N) = aa_nat_nat(modulo_modulo_nat(Y2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),X))
       => ? [Q3: nat] : X = aa_nat_nat(plus_plus_nat(Y2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_770_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_771_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_772_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M6: nat] : N = aa_nat_nat(suc,M6) ) ).

% gr0_conv_Suc
tff(fact_773_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_774_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% gr0_implies_Suc
tff(fact_775_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_776_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_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_777_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_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_778_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))
            & ! [I: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
               => ~ pp(aa_nat_bool(P,I)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_779_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
          & ( aa_nat_nat(plus_plus_nat(I2),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_780_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_781_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_782_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_783_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_784_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),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),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_785_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_786_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_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_787_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(power_power_nat(I2),M)),aa_nat_nat(power_power_nat(I2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_788_real__arch__pow__inv,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N3)),Y2)) ) ) ).

% real_arch_pow_inv
tff(fact_789_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_790_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one) = X ).

% pow.simps(1)
tff(fact_791_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(modulo_modulo_nat(aa_nat_nat(divide_divide_nat(M),N)),Q2))),aa_nat_nat(modulo_modulo_nat(M),N)) ).

% mod_mult2_eq
tff(fact_792_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M7: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X3: nat] :
            ( pp(aa_nat_bool(P,X3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M7)) )
       => ~ ! [M5: nat] :
              ( pp(aa_nat_bool(P,M5))
             => ~ ! [X2: nat] :
                    ( pp(aa_nat_bool(P,X2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_793_numeral__1__eq__Suc__0,axiom,
    numeral_numeral_nat(one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_794_num_Osize_I5_J,axiom,
    ! [X23: num] : size_size_num(aa_num_num(bit0,X23)) = aa_nat_nat(plus_plus_nat(size_size_num(X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_795_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))
            & ! [I: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
               => ~ pp(aa_nat_bool(P,I)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_796_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_797_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_798_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_799_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))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(P,N3))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_800_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_801_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(power_power_nat(N),K))) ) ).

% power_gt_expt
tff(fact_802_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(divide_divide_nat(K),N)),aa_nat_nat(divide_divide_nat(K),M))) ) ) ).

% div_le_mono2
tff(fact_803_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(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_804_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(power_power_nat(I2),N))) ) ).

% nat_one_le_power
tff(fact_805_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(divide_divide_nat(M),Q2)),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),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_806_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(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(divide_divide_nat(M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_807_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(divide_divide_nat(M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_808_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(divide_divide_nat(M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_809_numeral__2__eq__2,axiom,
    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_810_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),numeral_numeral_nat(aa_num_num(bit0,one)))) ).

% pos2
tff(fact_811_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(divide_divide_nat(N),Q2)))
      <=> 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),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_812_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(divide_divide_nat(M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_813_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(plus_plus_nat(N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_814_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(plus_plus_nat(N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(divide_divide_nat(M),N))))) ) ).

% dividend_less_times_div
tff(fact_815_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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_816_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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_817_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))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P,N3))
             => pp(aa_nat_bool(P,aa_nat_nat(plus_plus_nat(N3),numeral_numeral_nat(aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_818_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(divide_divide_nat(M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q4: 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))))
            & pp(aa_nat_bool(P,Q4)) ) ) ) ).

% split_div'
tff(fact_819_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_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(vEBT_VEBT_high(X),N)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_820_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_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(vEBT_VEBT_low(X),N)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_821_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(plus_plus_nat(N),M) )
             => ( ! [I3: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M)))
                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_1))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(vEBT_VEBT_low(Ma),N))) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(vEBT_VEBT_high(X3),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(vEBT_VEBT_low(X3),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),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(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_822_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N3))) ) )
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P,N3))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N3)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_823_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(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_824_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(divide_divide_nat(aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_825_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(plus_plus_nat(N),M) )
             => ( ! [I3: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M)))
                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_1))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(vEBT_VEBT_low(Ma),N))) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(vEBT_VEBT_high(X3),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(vEBT_VEBT_low(X3),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),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(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_826_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(plus_plus_nat(aa_nat_nat(divide_divide_nat(A2),N)),aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(modulo_modulo_nat(B2),N)),zero_zero_nat),one_one_nat),zero_zero_nat))),aa_nat_nat(divide_divide_nat(B2),N))) ) ) ).

% verit_le_mono_div
tff(fact_827_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_828_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_829_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] : A2 = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(A2),N)),N)),aa_nat_nat(modulo_modulo_nat(A2),N)) ).

% div_mod_decomp
tff(fact_830_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y23: num] :
      ( ( aa_num_num(bit0,X23) = aa_num_num(bit0,Y23) )
    <=> ( X23 = Y23 ) ) ).

% verit_eq_simplify(8)
tff(fact_831_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_832_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_833_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),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_834_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),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_835_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),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_836_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_837_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_838_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_839_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))
       => ( modulo_modulo_int(K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_840_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))
       => ( modulo_modulo_int(K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_841_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),modulo_modulo_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_842_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),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_843_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))
       => ( divide_divide_int(K,L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_844_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))
       => ( divide_divide_int(K,L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_845_times__int__code_I2_J,axiom,
    ! [L: int] : times_times_int(zero_zero_int,L) = zero_zero_int ).

% times_int_code(2)
tff(fact_846_times__int__code_I1_J,axiom,
    ! [K: int] : times_times_int(K,zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_847_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo_int(M,D) = zero_zero_int )
     => ? [Q3: int] : M = times_times_int(D,Q3) ) ).

% zmod_eq_0D
tff(fact_848_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo_int(M,D) = zero_zero_int )
    <=> ? [Q4: int] : M = times_times_int(D,Q4) ) ).

% zmod_eq_0_iff
tff(fact_849_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_850_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,plus_plus_int(K,L)),zero_zero_int))
       => ( modulo_modulo_int(K,L) = plus_plus_int(K,L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_851_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))
     => ( modulo_modulo_int(A,times_times_int(B,C)) = plus_plus_int(times_times_int(B,modulo_modulo_int(divide_divide_int(A,B),C)),modulo_modulo_int(A,B)) ) ) ).

% zmod_zmult2_eq
tff(fact_852_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))
     => ( divide_divide_int(A,times_times_int(B,C)) = divide_divide_int(divide_divide_int(A,B),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_853_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,plus_plus_int(times_times_int(B,Q5),R4)),plus_plus_int(times_times_int(B,Q2),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),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_854_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),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_855_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),modulo_modulo_int(K,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_856_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,modulo_modulo_int(K,L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_857_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,modulo_modulo_int(M,K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_858_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),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_859_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),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_860_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,plus_plus_int(times_times_int(B,Q5),R4)),plus_plus_int(times_times_int(B,Q2),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),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),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_861_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_862_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( plus_plus_int(times_times_int(B,Q2),R2) = plus_plus_int(times_times_int(B3,Q5),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,plus_plus_int(times_times_int(B3,Q5),R4)),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),R4))
           => ( 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(ord_less_eq_int,B3),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_863_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: 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),divide_divide_int(I2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_864_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,divide_divide_int(A,B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_865_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,divide_divide_int(A,B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_866_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,plus_plus_int(divide_divide_int(A2,N),if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,modulo_modulo_int(B2,N)),zero_zero_int),one_one_int,zero_zero_int))),divide_divide_int(B2,N))) ) ) ).

% verit_le_mono_div_int
tff(fact_867_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,divide_divide_int(X,K)),X)) ) ) ).

% int_div_less_self
tff(fact_868_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( modulo_modulo_int(I2,K) = I2 )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_869_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( plus_plus_int(times_times_int(B,Q2),R2) = plus_plus_int(times_times_int(B3,Q5),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B3,Q5),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
         => ( 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),B3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_870_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),divide_divide_int(K,L))) ) ) ).

% div_positive_int
tff(fact_871_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,divide_divide_int(N,K)),modulo_modulo_int(N,K)))
      <=> ! [I4: 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 = plus_plus_int(times_times_int(K,I4),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_872_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,divide_divide_int(N,K)),modulo_modulo_int(N,K)))
      <=> ! [I4: 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 = plus_plus_int(times_times_int(K,I4),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_873_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),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_874_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] : A2 = plus_plus_int(times_times_int(divide_divide_int(A2,N),N),modulo_modulo_int(A2,N)) ).

% div_mod_decomp_int
tff(fact_875_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,plus_plus_int(W,one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_876_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))
     => ( ( times_times_int(M,N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_877_zdiv__mono2__neg,axiom,
    ! [A: int,B3: 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),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,divide_divide_int(A,B3)),divide_divide_int(A,B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_878_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
     => ( 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,divide_divide_int(A3,B)),divide_divide_int(A,B))) ) ) ).

% zdiv_mono1_neg
tff(fact_879_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = plus_plus_int(times_times_int(B,Q2),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))
         => ( modulo_modulo_int(A,B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_880_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = plus_plus_int(times_times_int(B,Q2),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))
         => ( modulo_modulo_int(A,B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_881_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = plus_plus_int(times_times_int(B,Q2),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))
         => ( divide_divide_int(A,B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_882_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = plus_plus_int(times_times_int(B,Q2),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))
         => ( divide_divide_int(A,B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_883_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),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,times_times_int(K,I2)),times_times_int(K,J))) ) ) ).

% zmult_zless_mono2
tff(fact_884_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( divide_divide_int(I2,K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_885_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),modulo_modulo_int(A,B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,modulo_modulo_int(A,B)),B)) ) ) ).

% pos_mod_conj
tff(fact_886_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,modulo_modulo_int(A,B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),modulo_modulo_int(A,B))) ) ) ).

% neg_mod_conj
tff(fact_887_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B3,Q5),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),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(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_888_zdiv__mono2,axiom,
    ! [A: int,B3: 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),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,divide_divide_int(A,B)),divide_divide_int(A,B3))) ) ) ) ).

% zdiv_mono2
tff(fact_889_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
     => ( 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,divide_divide_int(A,B)),divide_divide_int(A3,B))) ) ) ).

% zdiv_mono1
tff(fact_890_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,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))
         => ! [I4: 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 = plus_plus_int(times_times_int(K,I4),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: 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 = plus_plus_int(times_times_int(K,I4),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_891_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,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))
         => ! [I4: 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 = plus_plus_int(times_times_int(K,I4),J3) ) )
             => pp(aa_int_bool(P,I4)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: 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 = plus_plus_int(times_times_int(K,I4),J3) ) )
             => pp(aa_int_bool(P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_892_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),plus_plus_int(one_one_int,Z))) ) ).

% le_imp_0_less
tff(fact_893_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_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_894_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_895_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_896_int__gr__induct,axiom,
    ! [K: int,I2: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
     => ( pp(aa_int_bool(P,plus_plus_int(K,one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,plus_plus_int(I3,one_one_int))) ) )
         => pp(aa_int_bool(P,I2)) ) ) ) ).

% int_gr_induct
tff(fact_897_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),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_898_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,plus_plus_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_899_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_900_odd__nonzero,axiom,
    ! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_901_plus__int__code_I1_J,axiom,
    ! [K: int] : plus_plus_int(K,zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_902_plus__int__code_I2_J,axiom,
    ! [L: int] : plus_plus_int(zero_zero_int,L) = L ).

% plus_int_code(2)
tff(fact_903_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,modulo_modulo_int(K,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_904_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),modulo_modulo_int(K,L))) ) ).

% neg_mod_bound
tff(fact_905_int__distrib_I2_J,axiom,
    ! [W: int,Z12: int,Z22: int] : times_times_int(W,plus_plus_int(Z12,Z22)) = plus_plus_int(times_times_int(W,Z12),times_times_int(W,Z22)) ).

% int_distrib(2)
tff(fact_906_int__distrib_I1_J,axiom,
    ! [Z12: int,Z22: int,W: int] : times_times_int(plus_plus_int(Z12,Z22),W) = plus_plus_int(times_times_int(Z12,W),times_times_int(Z22,W)) ).

% int_distrib(1)
tff(fact_907_int__ge__induct,axiom,
    ! [K: int,I2: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,plus_plus_int(I3,one_one_int))) ) )
         => pp(aa_int_bool(P,I2)) ) ) ) ).

% int_ge_induct
tff(fact_908_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_909_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,bit_se4203085406695923979it_int(N,K)),K)) ).

% unset_bit_less_eq
tff(fact_910_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),bit_se7879613467334960850it_int(N,K))) ).

% set_bit_greater_eq
tff(fact_911_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))
     => ( modulo_modulo_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),B)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),A)) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),modulo_modulo_int(B,A))) ) ) ).

% pos_zmod_mult_2
tff(fact_912_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one != aa_num_num(bit0,X23) ).

% verit_eq_simplify(10)
tff(fact_913_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(modulo_modulo_nat(A2),N) = zero_zero_nat )
         => ( ( aa_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(divide_divide_nat(A2),N)),aa_nat_nat(divide_divide_nat(B2),N))) ) ) ) ) ).

% div_less_mono
tff(fact_914_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_915_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_916_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,plus_plus_int(X3,D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X2: int] :
              ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(P,plus_plus_int(X2,times_times_int(K,D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_917_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),X) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_918_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_919_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: 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(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(plus_plus_nat(N),M) )
             => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_920_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S),X) = vEBT_Node(Info,zero_zero_nat,Ts,S) ).

% vebt_insert.simps(2)
tff(fact_921_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_922_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_923_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A5: bool,B5: bool] : T = vEBT_Leaf(A5,B5) ) ).

% deg_1_Leaf
tff(fact_924_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A5: bool,B5: bool] : T = vEBT_Leaf(A5,B5) ) ) ).

% deg_1_Leafy
tff(fact_925_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,Uv: bool,D2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_926_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv: bool] : X != vEBT_Leaf(fTrue,Uv)
       => ( ! [Uu: bool] : X != vEBT_Leaf(Uu,fTrue)
         => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,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_927_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_928_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y2 != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X222: bool] : Y2 != vEBT_Leaf(X212,X222) ) ).

% VEBT.exhaust
tff(fact_929_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_930_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_931_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_932_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv2))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_933_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu2: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu2,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_934_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: bool,B5: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),X3)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S2),X3)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),X3)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_935_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: bool,B5: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),X3)
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Ux)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_936_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,Uv: bool,Uw: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Uw)
     => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Uz2)
       => ( ! [Mi2: 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(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),X3)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_937_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: bool,B5: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),X3)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),X3)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X3)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_938_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y2: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y2) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y2) )
       => ( ( ? [Uv: bool] : X = vEBT_Leaf(fTrue,Uv)
           => pp(Y2) )
         => ( ( ? [Uu: bool] : X = vEBT_Leaf(Uu,fTrue)
             => pp(Y2) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy2)
               => ~ pp(Y2) )
             => ~ ( ? [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(Y2) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_939_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: bool,B: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(fTrue,B) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_940_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),X)) ).

% vebt_member.simps(2)
tff(fact_941_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_942_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_943_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_944_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_945_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Ux2)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_946_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv: bool] : X != vEBT_Leaf(fTrue,Uv)
       => ( ! [Uu: bool] : X != vEBT_Leaf(Uu,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_947_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_948_imp__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( 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),X5))
           => pp(P3) ) ) ) ) ).

% imp_le_cong
tff(fact_949_conj__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( 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),X5))
            & pp(P3) ) ) ) ) ).

% conj_le_cong
tff(fact_950_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A5: bool,B5: bool] : A1 = vEBT_Leaf(A5,B5)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X2),N3)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5) )
                     => ( ( M5 = N3 )
                       => ( ( Deg2 = aa_nat_nat(plus_plus_nat(N3),M5) )
                         => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X2),N3)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5) )
                       => ( ( M5 = aa_nat_nat(suc,N3) )
                         => ( ( Deg2 = aa_nat_nat(plus_plus_nat(N3),M5) )
                           => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X2),N3)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5) )
                         => ( ( M5 = N3 )
                           => ( ( Deg2 = aa_nat_nat(plus_plus_nat(N3),M5) )
                             => ( ! [I: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5)))
                                   => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),X_1))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5)))
                                             => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma2),N3) = I )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),aa_nat_nat(vEBT_VEBT_low(Ma2),N3))) )
                                                & ! [X2: nat] :
                                                    ( ( ( aa_nat_nat(vEBT_VEBT_high(X2),N3) = I )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),aa_nat_nat(vEBT_VEBT_low(X2),N3))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X2))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X2),N3)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5) )
                           => ( ( M5 = aa_nat_nat(suc,N3) )
                             => ( ( Deg2 = aa_nat_nat(plus_plus_nat(N3),M5) )
                               => ( ! [I: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5)))
                                     => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),X_1))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M5)))
                                               => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma2),N3) = I )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),aa_nat_nat(vEBT_VEBT_low(Ma2),N3))) )
                                                  & ! [X2: nat] :
                                                      ( ( ( aa_nat_nat(vEBT_VEBT_high(X2),N3) = I )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),aa_nat_nat(vEBT_VEBT_low(X2),N3))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X2))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_951_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) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(plus_plus_nat(N4),N4) )
            & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_1))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(plus_plus_nat(N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_1))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
        | ? [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) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(plus_plus_nat(N4),N4) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N4)))
               => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_1))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [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,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N4)))
                 => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma3),N4) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(vEBT_VEBT_low(Ma3),N4))) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(vEBT_VEBT_high(X4),N4) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(vEBT_VEBT_low(X4),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X4))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),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) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(plus_plus_nat(N4),aa_nat_nat(suc,N4)) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,N4))))
               => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_1))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [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,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,N4))))
                 => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma3),N4) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(vEBT_VEBT_low(Ma3),N4))) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(vEBT_VEBT_high(X4),N4) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(vEBT_VEBT_low(X4),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X4))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_952_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: 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(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(plus_plus_nat(N),M) )
             => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_953_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_954_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_955_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: 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,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),A)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),B),product_Pair_int_int(Q2,plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_956_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),zero_zero_nat))
     => ( ! [M5: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N3),aa_nat_nat(modulo_modulo_nat(M5),N3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),N3)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_957_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N),K),L) = plus_plus_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),aa_int_int(bit_concat_bit(N,divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_958_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_959_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_960_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_961_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_962_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(plus_plus_nat(M),N),aa_int_int(bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_963_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = times_times_int(Q2,L) )
       => eucl_rel_int(K,L,product_Pair_int_int(Q2,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_964_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_965_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: 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(Mi),Ma)),zero_zero_nat,Va2,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_966_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A5),B5))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B5),A5)) )
     => ( ! [A5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A5),zero_zero_nat))
       => ( ! [A5: nat,B5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A5),B5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A5),aa_nat_nat(plus_plus_nat(A5),B5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_967_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R2))
    <=> ( ( K = plus_plus_int(times_times_int(L,Q2),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))
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_968_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( eucl_rel_int(plus_plus_int(A,one_one_int),B,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),A)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),B),product_Pair_int_int(Q2,minus_minus_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_969_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y2 )
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y2 = vEBT_Leaf(fTrue,B5) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y2 = vEBT_Leaf(A5,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y2 = vEBT_Leaf(A5,B5) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) )
             => ( Y2 != vEBT_Node(Info2,zero_zero_nat,Ts2,S2) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
               => ( Y2 != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                 => ( Y2 != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                   => ( Y2 != if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(vEBT_VEBT_low(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_970_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_971_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_972_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_973_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_974_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_975_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_976_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_977_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),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_978_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_979_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_980_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = times_times_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_981_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : bit_ri631733984087533419it_int(N,minus_minus_int(bit_ri631733984087533419it_int(N,K),bit_ri631733984087533419it_int(N,L))) = bit_ri631733984087533419it_int(N,minus_minus_int(K,L)) ).

% signed_take_bit_diff
tff(fact_982_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : bit_ri631733984087533419it_int(N,times_times_int(bit_ri631733984087533419it_int(N,K),bit_ri631733984087533419it_int(N,L))) = bit_ri631733984087533419it_int(N,times_times_int(K,L)) ).

% signed_take_bit_mult
tff(fact_983_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : bit_ri631733984087533419it_int(N,plus_plus_int(bit_ri631733984087533419it_int(N,K),bit_ri631733984087533419it_int(N,L))) = bit_ri631733984087533419it_int(N,plus_plus_int(K,L)) ).

% signed_take_bit_add
tff(fact_984_int__distrib_I4_J,axiom,
    ! [W: int,Z12: int,Z22: int] : times_times_int(W,minus_minus_int(Z12,Z22)) = minus_minus_int(times_times_int(W,Z12),times_times_int(W,Z22)) ).

% int_distrib(4)
tff(fact_985_int__distrib_I3_J,axiom,
    ! [Z12: int,Z22: int,W: int] : times_times_int(minus_minus_int(Z12,Z22),W) = minus_minus_int(times_times_int(Z12,W),times_times_int(Z22,W)) ).

% int_distrib(3)
tff(fact_986_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_987_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))),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))),plus_plus_int(K,L))) ) ).

% even_diff_iff
tff(fact_988_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_989_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_990_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_991_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,times_times_int(K,M)),times_times_int(K,T))) ) ) ).

% zdvd_mono
tff(fact_992_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,times_times_int(K,M)),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_993_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_994_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y2: 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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),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(plus_plus_nat(A),B)))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A),B)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A),B)),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_995_int__le__induct,axiom,
    ! [I2: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,minus_minus_int(I3,one_one_int))) ) )
         => pp(aa_int_bool(P,I2)) ) ) ) ).

% int_le_induct
tff(fact_996_int__less__induct,axiom,
    ! [I2: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
     => ( pp(aa_int_bool(P,minus_minus_int(K,one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,minus_minus_int(I3,one_one_int))) ) )
         => pp(aa_int_bool(P,I2)) ) ) ) ).

% int_less_induct
tff(fact_997_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),plus_plus_int(N,times_times_int(K,M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_998_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),plus_plus_int(X,T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),plus_plus_int(plus_plus_int(X,times_times_int(C,D)),T))) ) ) ).

% zdvd_period
tff(fact_999_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,bit_ri631733984087533419it_int(N,K)),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_1000_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_1001_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_1002_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_1003_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1004_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_1005_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(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_1006_plusinfinity,axiom,
    ! [D: int,P3: 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(P3,X3))
          <=> pp(aa_int_bool(P3,minus_minus_int(X3,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(P3,X3)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P3,X_12))
           => ? [X_13: int] : pp(aa_int_bool(P,X_13)) ) ) ) ) ).

% plusinfinity
tff(fact_1007_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,minus_minus_int(X3,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_12: int] : pp(aa_int_bool(P1,X_12))
           => ? [X_13: int] : pp(aa_int_bool(P,X_13)) ) ) ) ) ).

% minusinfinity
tff(fact_1008_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I2: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
           => ( pp(aa_int_bool(P,I3))
             => pp(aa_int_bool(P,plus_plus_int(I3,one_one_int))) ) )
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,minus_minus_int(I3,one_one_int))) ) )
         => pp(aa_int_bool(P,I2)) ) ) ) ).

% int_induct
tff(fact_1009_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_1010_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_1011_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_1012_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(power_power_nat(I2),M)),aa_nat_nat(power_power_nat(I2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_1013_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,minus_minus_int(X3,D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X2: int] :
              ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(P,minus_minus_int(X2,times_times_int(K,D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1014_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),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_1015_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))
       => ( modulo_modulo_int(K,L) = modulo_modulo_int(minus_minus_int(K,L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1016_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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_1017_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(power_power_nat(K),M)),aa_nat_nat(power_power_nat(K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_1018_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_1019_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),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(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_1020_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),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_1021_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_1022_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: 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),TreeList,Vd2)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),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(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_1023_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))
       => ( divide_divide_int(K,L) = plus_plus_int(divide_divide_int(minus_minus_int(K,L),L),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_1024_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ 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(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_1025_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: 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(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),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(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(X),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_1026_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A5) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B5) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1027_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A5) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B5) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1028_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y2) )
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ( pp(Y2)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B5) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
           => pp(Y2) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2)
               => ( pp(Y2)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1029_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_1030_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A5) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B5) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1031_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Uu: bool,Uv: bool] : X != vEBT_Leaf(Uu,Uv)
       => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1032_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y2) )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => pp(Y2) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)
           => pp(Y2) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y2)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( pp(Y2)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
                   => ( pp(Y2)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1033_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X) = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(vEBT_VEBT_low(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi),Mi),X)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)) ).

% vebt_insert.simps(5)
tff(fact_1034_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A5) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B5) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1035_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y2) )
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ( pp(Y2)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B5) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => pp(Y2) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => pp(Y2) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y2) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
                   => ( pp(Y2)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1036_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))
     => ( modulo_modulo_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),B)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),A)) = minus_minus_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),modulo_modulo_int(plus_plus_int(B,one_one_int),A)),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_1037_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y2 )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y2 = vEBT_Leaf(fTrue,B5) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y2 = vEBT_Leaf(A5,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y2 = vEBT_Leaf(A5,B5) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) )
               => ( ( Y2 = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S2),Xa2)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                 => ( ( Y2 = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),Xa2)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                   => ( ( Y2 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( ( Y2 = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(vEBT_VEBT_low(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2),Mi2),Xa2)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1038_div2__even__ext__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa_nat_nat(divide_divide_nat(X),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(divide_divide_nat(Y2),numeral_numeral_nat(aa_num_num(bit0,one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Y2)) )
       => ( X = Y2 ) ) ) ).

% div2_even_ext_nat
tff(fact_1039_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y2: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y2 )
     => ( ( ( X = zero_zero_nat )
         => ( Y2 != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y2 != 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,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y2 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y2 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1040_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y2) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( ( pp(Y2)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B5) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ~ pp(Y2)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y2)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(Y2)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( ( pp(Y2)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1041_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B5) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1042_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B5) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1043_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(minus_minus_nat(M),N) ).

% diff_Suc_Suc
tff(fact_1044_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(M),N)),K) ).

% Suc_diff_diff
tff(fact_1045_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(minus_minus_nat(zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1046_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1047_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
     => ( aa_nat_nat(minus_minus_nat(N),aa_nat_nat(minus_minus_nat(N),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_1048_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(I2),J)),K) = aa_nat_nat(minus_minus_nat(I2),aa_nat_nat(plus_plus_nat(J),K)) ).

% diff_diff_left
tff(fact_1049_idiff__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_1050_idiff__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_1051_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(minus_minus_nat(N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_1052_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(minus_minus_nat(M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1053_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_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_1054_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(minus_minus_nat(I2),aa_nat_nat(minus_minus_nat(J),K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(I2),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1055_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(J),K)),I2) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(J),I2)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1056_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(plus_plus_nat(I2),aa_nat_nat(minus_minus_nat(J),K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(I2),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1057_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_1058_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(minus_minus_nat(N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_1059_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(J),K))),I2) = aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,J)),aa_nat_nat(plus_plus_nat(K),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1060_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(minus_minus_nat(I2),aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(J),K))) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(I2),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1061_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(minus_minus_nat(N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_1062_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_1063_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_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,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(plus_plus_nat(M),N))) ) ) ).

% even_diff_nat
tff(fact_1064_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_nat_nat(minus_minus_nat(N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1065_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_1066_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(minus_minus_nat(M),N))) ) ) ).

% dvd_diff_nat
tff(fact_1067_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I2: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P,N3)) )
       => pp(aa_nat_bool(P,aa_nat_nat(minus_minus_nat(K),I2))) ) ) ).

% zero_induct_lemma
tff(fact_1068_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_1069_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(minus_minus_nat(M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(minus_minus_nat(N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1070_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(minus_minus_nat(J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_1071_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(minus_minus_nat(L),N)),aa_nat_nat(minus_minus_nat(L),M))) ) ) ).

% diff_less_mono2
tff(fact_1072_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_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_1073_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(minus_minus_nat(L),N)),aa_nat_nat(minus_minus_nat(L),M))) ) ).

% diff_le_mono2
tff(fact_1074_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(minus_minus_nat(C),A)),aa_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_1075_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(M),N)),M)) ).

% diff_le_self
tff(fact_1076_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(minus_minus_nat(M),L)),aa_nat_nat(minus_minus_nat(N),L))) ) ).

% diff_le_mono
tff(fact_1077_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(minus_minus_nat(aa_nat_nat(minus_minus_nat(M),K)),aa_nat_nat(minus_minus_nat(N),K)) = aa_nat_nat(minus_minus_nat(M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1078_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(minus_minus_nat(M),K)),aa_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_1079_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(minus_minus_nat(M),K) = aa_nat_nat(minus_minus_nat(N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1080_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(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_1081_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(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_1082_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(minus_minus_nat(N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_1083_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(M),N)),N) = M ).

% diff_add_inverse2
tff(fact_1084_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(N),M)),N) = M ).

% diff_add_inverse
tff(fact_1085_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(M),K)),aa_nat_nat(plus_plus_nat(N),K)) = aa_nat_nat(minus_minus_nat(M),N) ).

% diff_cancel2
tff(fact_1086_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(K),M)),aa_nat_nat(plus_plus_nat(K),N)) = aa_nat_nat(minus_minus_nat(M),N) ).

% Nat.diff_cancel
tff(fact_1087_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(M),N)),K) = aa_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_1088_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(minus_minus_nat(M),N)) = aa_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_1089_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
      & ( ( aa_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)) = D2 )
        | ( aa_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)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_1090_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(minus_minus_nat(M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_1091_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(minus_minus_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(minus_minus_nat(M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1092_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(minus_minus_nat(M),N)),M)) ) ) ).

% diff_less
tff(fact_1093_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(minus_minus_nat(aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(M),N)) ) ) ).

% Suc_diff_le
tff(fact_1094_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(minus_minus_nat(A),C)),aa_nat_nat(minus_minus_nat(B),C))) ) ) ).

% diff_less_mono
tff(fact_1095_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(minus_minus_nat(M),K)),aa_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_1096_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(minus_minus_nat(N),aa_nat_nat(plus_plus_nat(N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_1097_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(plus_plus_nat(N),aa_nat_nat(minus_minus_nat(M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1098_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(minus_minus_nat(J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K)),J)) ) ).

% less_diff_conv
tff(fact_1099_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( ( aa_nat_nat(minus_minus_nat(J),I2) = K )
      <=> ( J = aa_nat_nat(plus_plus_nat(K),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1100_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(J),I2)),K) = aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(J),K)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1101_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(I2),J)),K) = aa_nat_nat(plus_plus_nat(I2),aa_nat_nat(minus_minus_nat(J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1102_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: 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,I2),aa_nat_nat(minus_minus_nat(J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(I2),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1103_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(J),K)),I2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(plus_plus_nat(I2),K))) ) ).

% le_diff_conv
tff(fact_1104_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,N)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1105_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_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(modulo_modulo_nat(M),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(minus_minus_nat(M),N)),N) ) ) ) ).

% mod_if
tff(fact_1106_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(modulo_modulo_nat(M),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(minus_minus_nat(M),N)),N) ) ) ).

% mod_geq
tff(fact_1107_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(modulo_modulo_nat(M),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(minus_minus_nat(M),N)),N) ) ) ).

% le_mod_geq
tff(fact_1108_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(modulo_modulo_nat(M),Q2) = aa_nat_nat(modulo_modulo_nat(N),Q2) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(minus_minus_nat(M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1109_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(plus_plus_nat(aa_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_1110_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y2: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y2))
     => ( plus_p3455044024723400733d_enat(X,minus_3235023915231533773d_enat(Y2,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y2),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1111_diff__Suc__less,axiom,
    ! [N: nat,I2: 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(minus_minus_nat(N),aa_nat_nat(suc,I2))),N)) ) ).

% diff_Suc_less
tff(fact_1112_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_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(plus_plus_nat(B),D3) )
              & ~ pp(aa_nat_bool(P,D3)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1113_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_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(plus_plus_nat(B),D3) )
           => pp(aa_nat_bool(P,D3)) ) ) ) ).

% nat_diff_split
tff(fact_1114_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: 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(minus_minus_nat(J),K)),I2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(plus_plus_nat(I2),K))) ) ) ).

% less_diff_conv2
tff(fact_1115_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1116_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1117_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1118_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1119_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1120_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1121_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),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(minus_minus_nat(N),Q2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(plus_plus_nat(N),aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_1122_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(minus_minus_nat(M),R2)))
         => ( aa_nat_nat(modulo_modulo_nat(M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1123_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(modulo_modulo_nat(M),N) = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M),N)),N)) ).

% modulo_nat_def
tff(fact_1124_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(minus_minus_nat(aa_nat_nat(suc,M)),N) = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(minus_minus_nat(N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1125_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(minus_minus_nat(N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1126_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(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(divide_divide_nat(M),N) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(M),N)),N)) ) ) ) ).

% div_if
tff(fact_1127_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(divide_divide_nat(M),N) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(M),N)),N)) ) ) ) ).

% div_geq
tff(fact_1128_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(plus_plus_nat(M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(plus_plus_nat(M),N) = aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_1129_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1130_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1131_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(plus_plus_nat(N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_1132_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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(minus_minus_nat(M),N)),aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(K),M)),aa_nat_nat(power_power_nat(K),N)))) ) ).

% diff_le_diff_pow
tff(fact_1133_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(divide_divide_nat(M),N) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_1134_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] : X != aa_nat_nat(suc,N3) ) ).

% list_decode.cases
tff(fact_1135_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(N),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,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_1136_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))
       => ( divide_divide_int(aa_nat_int(power_power_int(K),M),K) = aa_nat_int(power_power_int(K),aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_1137_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_1138_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1139_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(B),A)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1140_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y2) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( ( pp(Y2)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B5) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ( ~ pp(Y2)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2) )
                 => ( ( pp(Y2)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1141_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B5) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2),Xa2))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1142_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A5,B5),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B5) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2),Xa2))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1143_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa2)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1144_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y2) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ~ pp(Y2)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa2)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2) )
               => ( ~ pp(Y2)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y2)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( ( pp(Y2)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( ( pp(Y2)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1145_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(I2),J)),K) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(I2),K)),J) ).

% diff_commute
tff(fact_1146_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),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(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa2),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,V2)),numeral_numeral_nat(aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1147_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat))) ) ).

% inrange
tff(fact_1148_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A5: real,B5: real,C2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A5),B5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B5),C2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A5),B5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B5),C2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A5),C2)) ) ) ) )
       => ( ! [X3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
               => ? [D4: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
                    & ! [A5: real,B5: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A5),X3))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B5))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(B5),A5)),D4)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A5),B5)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_1149_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_1150_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y2: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y2 )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y2 = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y2 = 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,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y2 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y2 = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),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_1151_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(plus_plus_real(X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X = A ) ) ).

% real_add_minus_iff
tff(fact_1152_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_1153_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : 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)))) = times_times_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_1154_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_ri631733984087533419it_int(N,aa_int_int(uminus_uminus_int,bit_ri631733984087533419it_int(N,K))) = bit_ri631733984087533419it_int(N,aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_1155_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N))
         => pp(aa_nat_bool(P,M6)) )
    <=> ! [X4: nat] :
          ( member_nat(X4,set_or1269000886237332187st_nat(zero_zero_nat,N))
         => pp(aa_nat_bool(P,X4)) ) ) ).

% all_nat_less
tff(fact_1156_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N))
          & pp(aa_nat_bool(P,M6)) )
    <=> ? [X4: nat] :
          ( member_nat(X4,set_or1269000886237332187st_nat(zero_zero_nat,N))
          & pp(aa_nat_bool(P,X4)) ) ) ).

% ex_nat_less
tff(fact_1157_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(times_times_real(U),U))),aa_real_real(times_times_real(X),X))) ).

% real_minus_mult_self_le
tff(fact_1158_zmult__eq__1__iff,axiom,
    ! [M: int,N: 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_1159_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: 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_1160_minus__real__def,axiom,
    ! [X: real,Y2: real] : aa_real_real(minus_minus_real(X),Y2) = aa_real_real(plus_plus_real(X),aa_real_real(uminus_uminus_real,Y2)) ).

% minus_real_def
tff(fact_1161_real__0__less__add__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(X),Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y2)) ) ).

% real_0_less_add_iff
tff(fact_1162_real__add__less__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(X),Y2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_1163_real__add__le__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(X),Y2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_le_0_iff
tff(fact_1164_real__0__le__add__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(X),Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y2)) ) ).

% real_0_le_add_iff
tff(fact_1165_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,divide_divide_int(B2,N)),divide_divide_int(A2,N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_1166_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( 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_1167_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(aa_real_fun_nat_real(power_power_real,U),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_1168_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))
     => ( modulo_modulo_int(aa_int_int(uminus_uminus_int,K),L) = minus_minus_int(minus_minus_int(L,one_one_int),modulo_modulo_int(minus_minus_int(K,one_one_int),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1169_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int),B) = minus_minus_int(B,one_one_int) ) ) ).

% zmod_minus1
tff(fact_1170_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( modulo_modulo_int(A,B) = zero_zero_int )
         => ( divide_divide_int(A,aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,divide_divide_int(A,B)) ) )
        & ( ( modulo_modulo_int(A,B) != zero_zero_int )
         => ( divide_divide_int(A,aa_int_int(uminus_uminus_int,B)) = minus_minus_int(aa_int_int(uminus_uminus_int,divide_divide_int(A,B)),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_1171_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( modulo_modulo_int(A,B) = zero_zero_int )
         => ( divide_divide_int(aa_int_int(uminus_uminus_int,A),B) = aa_int_int(uminus_uminus_int,divide_divide_int(A,B)) ) )
        & ( ( modulo_modulo_int(A,B) != zero_zero_int )
         => ( divide_divide_int(aa_int_int(uminus_uminus_int,A),B) = minus_minus_int(aa_int_int(uminus_uminus_int,divide_divide_int(A,B)),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_1172_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,product_Pair_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),aa_int_int(uminus_uminus_int,Q2),minus_minus_int(aa_int_int(uminus_uminus_int,Q2),one_one_int)),if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),zero_zero_int,minus_minus_int(B,R2)))) ) ) ).

% zminus1_lemma
tff(fact_1173_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : 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_1174_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,plus_plus_int(K,L)),zero_zero_int))
       => ( divide_divide_int(K,L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1175_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,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_1176_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))),bit_ri631733984087533419it_int(N,K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_1177_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),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_1178_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,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,plus_plus_int(one_one_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_1179_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)))
       => ( bit_ri631733984087533419it_int(N,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_1180_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: 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_1181_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,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)))),bit_ri631733984087533419it_int(N,K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_1182_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : 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)))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(N,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_1183_set__decode__0,axiom,
    ! [X: nat] :
      ( member_nat(zero_zero_nat,nat_set_decode(X))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_1184_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( member_nat(aa_nat_nat(suc,N),nat_set_decode(X))
    <=> member_nat(N,nat_set_decode(aa_nat_nat(divide_divide_nat(X),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_1185_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_1186_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).

% semiring_norm(89)
tff(fact_1187_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_1188_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_1189_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_1190_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(73)
tff(fact_1191_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(80)
tff(fact_1192_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,zero_zero_nat)),N) = 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_1193_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : plus_plus_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_num_num(bit1,plus_plus_num(M,N)) ).

% semiring_norm(7)
tff(fact_1194_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : plus_plus_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_num_num(bit1,plus_plus_num(M,N)) ).

% semiring_norm(9)
tff(fact_1195_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : times_times_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_num_num(bit0,times_times_num(M,aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_1196_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : times_times_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_num_num(bit0,times_times_num(aa_num_num(bit1,M),N)) ).

% semiring_norm(15)
tff(fact_1197_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(72)
tff(fact_1198_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(81)
tff(fact_1199_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_1200_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(ord_less_num(one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_1201_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = divide_divide_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_1202_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : plus_plus_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_num_num(bit0,plus_plus_num(plus_plus_num(M,N),one)) ).

% semiring_norm(10)
tff(fact_1203_semiring__norm_I8_J,axiom,
    ! [M: num] : plus_plus_num(aa_num_num(bit1,M),one) = aa_num_num(bit0,plus_plus_num(M,one)) ).

% semiring_norm(8)
tff(fact_1204_semiring__norm_I5_J,axiom,
    ! [M: num] : plus_plus_num(aa_num_num(bit0,M),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_1205_semiring__norm_I4_J,axiom,
    ! [N: num] : plus_plus_num(one,aa_num_num(bit1,N)) = aa_num_num(bit0,plus_plus_num(N,one)) ).

% semiring_norm(4)
tff(fact_1206_semiring__norm_I3_J,axiom,
    ! [N: num] : plus_plus_num(one,aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_1207_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : times_times_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_num_num(bit1,plus_plus_num(plus_plus_num(M,N),aa_num_num(bit0,times_times_num(M,N)))) ).

% semiring_norm(16)
tff(fact_1208_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(ord_less_num(M),N)) ) ).

% semiring_norm(74)
tff(fact_1209_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(ord_less_eq_num(M),N)) ) ).

% semiring_norm(79)
tff(fact_1210_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(divide_divide_nat(M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(divide_divide_nat(M),aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_1211_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),numeral_numeral_nat(V)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),M)),numeral_numeral_nat(V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1212_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_1213_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),numeral_numeral_nat(V)) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),M)),numeral_numeral_nat(V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1214_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : 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))) = plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),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_1215_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,N),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = plus_plus_int(times_times_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_1216_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X32: num] : aa_num_num(bit0,X23) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_1217_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_1218_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N3: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N3))
       => ( ! [N3: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N3))
         => ( ! [M5: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),one)
           => ( ! [M5: num,N3: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N3))
             => ( ! [M5: num,N3: num] : X != product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N3))
               => ( ! [M5: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),one)
                 => ( ! [M5: num,N3: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N3))
                   => ~ ! [M5: num,N3: num] : X != product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N3)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1219_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one )
     => ( ! [X24: num] : Y2 != aa_num_num(bit0,X24)
       => ~ ! [X33: num] : Y2 != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_1220_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : numeral_numeral_nat(aa_num_num(bit1,N)) = aa_nat_nat(suc,numeral_numeral_nat(aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_1221_bset_I1_J,axiom,
    ! [D5: 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,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,B2)
                 => ( X3 != plus_plus_int(Xb,Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,minus_minus_int(X3,D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X3 != plus_plus_int(Xb,Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,minus_minus_int(X3,D5))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,minus_minus_int(X2,D5)))
                & pp(aa_int_bool(Q,minus_minus_int(X2,D5))) ) ) ) ) ) ).

% bset(1)
tff(fact_1222_bset_I2_J,axiom,
    ! [D5: 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,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,B2)
                 => ( X3 != plus_plus_int(Xb,Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,minus_minus_int(X3,D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,B2)
                   => ( X3 != plus_plus_int(Xb,Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,minus_minus_int(X3,D5))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,minus_minus_int(X2,D5)))
                | pp(aa_int_bool(Q,minus_minus_int(X2,D5))) ) ) ) ) ) ).

% bset(2)
tff(fact_1223_aset_I1_J,axiom,
    ! [D5: 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,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,A2)
                 => ( X3 != minus_minus_int(Xb,Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,plus_plus_int(X3,D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X3 != minus_minus_int(Xb,Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,plus_plus_int(X3,D5))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,plus_plus_int(X2,D5)))
                & pp(aa_int_bool(Q,plus_plus_int(X2,D5))) ) ) ) ) ) ).

% aset(1)
tff(fact_1224_aset_I2_J,axiom,
    ! [D5: 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,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,A2)
                 => ( X3 != minus_minus_int(Xb,Xa) ) ) )
         => ( pp(aa_int_bool(P,X3))
           => pp(aa_int_bool(P,plus_plus_int(X3,D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,A2)
                   => ( X3 != minus_minus_int(Xb,Xa) ) ) )
           => ( pp(aa_int_bool(Q,X3))
             => pp(aa_int_bool(Q,plus_plus_int(X3,D5))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,plus_plus_int(X2,D5)))
                | pp(aa_int_bool(Q,plus_plus_int(X2,D5))) ) ) ) ) ) ).

% aset(2)
tff(fact_1225_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ord_less_eq_set_nat(nat_set_decode(M),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_1226_numeral__3__eq__3,axiom,
    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_1227_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_1228_bset_I9_J,axiom,
    ! [D: int,D5: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(X2,T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(minus_minus_int(X2,D5),T))) ) ) ) ).

% bset(9)
tff(fact_1229_bset_I10_J,axiom,
    ! [D: int,D5: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(X2,T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(minus_minus_int(X2,D5),T))) ) ) ) ).

% bset(10)
tff(fact_1230_aset_I9_J,axiom,
    ! [D: int,D5: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(X2,T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(plus_plus_int(X2,D5),T))) ) ) ) ).

% aset(9)
tff(fact_1231_aset_I10_J,axiom,
    ! [D: int,D5: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(X2,T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),plus_plus_int(plus_plus_int(X2,D5),T))) ) ) ) ).

% aset(10)
tff(fact_1232_num_Osize_I6_J,axiom,
    ! [X32: num] : size_size_num(aa_num_num(bit1,X32)) = aa_nat_nat(plus_plus_nat(size_size_num(X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_1233_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,minus_minus_int(X3,times_times_int(K2,D)))) )
       => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
        <=> ? [X4: int] :
              ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D))
              & pp(aa_int_bool(P,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1234_bset_I3_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(minus_minus_int(T,one_one_int),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( X2 = T )
             => ( minus_minus_int(X2,D5) = T ) ) ) ) ) ).

% bset(3)
tff(fact_1235_bset_I4_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(T,B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( X2 != T )
             => ( minus_minus_int(X2,D5) != T ) ) ) ) ) ).

% bset(4)
tff(fact_1236_bset_I5_J,axiom,
    ! [D5: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,minus_minus_int(X2,D5)),T)) ) ) ) ).

% bset(5)
tff(fact_1237_bset_I7_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(T,B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),minus_minus_int(X2,D5))) ) ) ) ) ).

% bset(7)
tff(fact_1238_aset_I3_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(plus_plus_int(T,one_one_int),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( X2 = T )
             => ( plus_plus_int(X2,D5) = T ) ) ) ) ) ).

% aset(3)
tff(fact_1239_aset_I4_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(T,A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( X2 != T )
             => ( plus_plus_int(X2,D5) != T ) ) ) ) ) ).

% aset(4)
tff(fact_1240_aset_I5_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(T,A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,plus_plus_int(X2,D5)),T)) ) ) ) ) ).

% aset(5)
tff(fact_1241_aset_I7_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),plus_plus_int(X2,D5))) ) ) ) ).

% aset(7)
tff(fact_1242_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(divide_divide_nat(aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_1243_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(plus_plus_nat(numeral_numeral_nat(aa_num_num(bit1,one))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_1244_aset_I8_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),plus_plus_int(X2,D5))) ) ) ) ).

% aset(8)
tff(fact_1245_aset_I6_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(plus_plus_int(T,one_one_int),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,plus_plus_int(X2,D5)),T)) ) ) ) ) ).

% aset(6)
tff(fact_1246_bset_I8_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( member_int(minus_minus_int(T,one_one_int),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),minus_minus_int(X2,D5))) ) ) ) ) ).

% bset(8)
tff(fact_1247_bset_I6_J,axiom,
    ! [D5: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X2 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,minus_minus_int(X2,D5)),T)) ) ) ) ).

% bset(6)
tff(fact_1248_cpmi,axiom,
    ! [D5: int,P: fun_int_bool,P3: fun_int_bool,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( ? [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(P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
                 => ! [Xb: int] :
                      ( member_int(Xb,B2)
                     => ( X3 != plus_plus_int(Xb,Xa) ) ) )
             => ( pp(aa_int_bool(P,X3))
               => pp(aa_int_bool(P,minus_minus_int(X3,D5))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa_int_bool(P3,X3))
              <=> pp(aa_int_bool(P3,minus_minus_int(X3,times_times_int(K2,D5)))) )
           => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
            <=> ( ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D5))
                    & pp(aa_int_bool(P3,X4)) )
                | ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D5))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,B2)
                        & pp(aa_int_bool(P,plus_plus_int(Xa4,X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1249_cppi,axiom,
    ! [D5: int,P: fun_int_bool,P3: fun_int_bool,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( ? [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(P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
                 => ! [Xb: int] :
                      ( member_int(Xb,A2)
                     => ( X3 != minus_minus_int(Xb,Xa) ) ) )
             => ( pp(aa_int_bool(P,X3))
               => pp(aa_int_bool(P,plus_plus_int(X3,D5))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa_int_bool(P3,X3))
              <=> pp(aa_int_bool(P3,minus_minus_int(X3,times_times_int(K2,D5)))) )
           => ( ? [X_1: int] : pp(aa_int_bool(P,X_1))
            <=> ( ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D5))
                    & pp(aa_int_bool(P3,X4)) )
                | ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D5))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,A2)
                        & pp(aa_int_bool(P,minus_minus_int(Xa4,X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1250_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_nat(aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_1251_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af,X)) ).

% set_decode_def
tff(fact_1252_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
      | ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_nat(aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_1253_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(pred_numeral(L),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_1254_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(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_1255_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_1256_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = numeral_numeral_nat(K) )
    <=> ( N = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_1257_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( numeral_numeral_nat(K) = aa_nat_nat(suc,N) )
    <=> ( pred_numeral(K) = N ) ) ).

% eq_numeral_Suc
tff(fact_1258_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,pred_numeral(K)),N)) ) ).

% less_numeral_Suc
tff(fact_1259_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)),numeral_numeral_nat(K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),pred_numeral(K))) ) ).

% less_Suc_numeral
tff(fact_1260_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa_num_num(bit1,K)) = numeral_numeral_nat(aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_1261_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,pred_numeral(K)),N)) ) ).

% le_numeral_Suc
tff(fact_1262_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)),numeral_numeral_nat(K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),pred_numeral(K))) ) ).

% le_Suc_numeral
tff(fact_1263_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(minus_minus_nat(numeral_numeral_nat(K)),aa_nat_nat(suc,N)) = aa_nat_nat(minus_minus_nat(pred_numeral(K)),N) ).

% diff_numeral_Suc
tff(fact_1264_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,N)),numeral_numeral_nat(K)) = aa_nat_nat(minus_minus_nat(N),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_1265_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_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,pred_numeral(K)),N)) ).

% max_numeral_Suc
tff(fact_1266_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_1267_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = times_times_int(bit_ri631733984087533419it_int(pred_numeral(L),aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_1268_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = times_times_int(bit_ri631733984087533419it_int(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_1269_numeral__eq__Suc,axiom,
    ! [K: num] : numeral_numeral_nat(K) = aa_nat_nat(suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_1270_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa_nat_nat(minus_minus_nat(numeral_numeral_nat(K)),one_one_nat) ).

% pred_numeral_def
tff(fact_1271_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)),modulo_modulo_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_1272_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(divide_divide_nat(numeral_numeral_nat(M)),numeral_numeral_nat(N))),aa_nat_nat(modulo_modulo_nat(numeral_numeral_nat(M)),numeral_numeral_nat(N))) ).

% divmod'_nat_def
tff(fact_1273_eq__diff__eq_H,axiom,
    ! [X: real,Y2: real,Z: real] :
      ( ( X = aa_real_real(minus_minus_real(Y2),Z) )
    <=> ( Y2 = aa_real_real(plus_plus_real(X),Z) ) ) ).

% eq_diff_eq'
tff(fact_1274_one__div__minus__numeral,axiom,
    ! [N: num] : 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_1275_minus__one__div__numeral,axiom,
    ! [N: num] : 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_1276_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : 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_1277_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : 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_1278_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(product_Pair_int_int(Q2,R2)) = plus_plus_int(Q2,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_1279_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(bitM(M),aa_num_num(bit0,one)) = product_Pair_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_1280_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_ag,set_or1266510415728281911st_int(M,N)) = divide_divide_int(minus_minus_int(times_times_int(N,plus_plus_int(N,one_one_int)),times_times_int(M,minus_minus_int(M,one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_1281_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(aa_num_num(bit0,K)) = numeral_numeral_nat(bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_1282_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_1283_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(aa_num_num(bit0,N)) = aa_num_num(bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_1284_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).

% semiring_norm(28)
tff(fact_1285_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,ring_1_of_int_real(divide_divide_int(N,X))),aa_real_real(divide_divide_real(ring_1_of_int_real(N)),ring_1_of_int_real(X)))) ).

% real_of_int_div4
tff(fact_1286_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),N))
     => ( ring_1_of_int_real(divide_divide_int(N,D)) = aa_real_real(divide_divide_real(ring_1_of_int_real(N)),ring_1_of_int_real(D)) ) ) ).

% real_of_int_div
tff(fact_1287_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : numeral_numeral_nat(aa_num_num(bit0,N)) = aa_nat_nat(suc,numeral_numeral_nat(bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_1288_one__plus__BitM,axiom,
    ! [N: num] : plus_plus_num(one,bitM(N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_1289_BitM__plus__one,axiom,
    ! [N: num] : plus_plus_num(bitM(N),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_1290_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,ring_1_of_int_real(N)),aa_real_real(plus_plus_real(ring_1_of_int_real(M)),one_one_real))) ) ).

% int_le_real_less
tff(fact_1291_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(plus_plus_real(ring_1_of_int_real(N)),one_one_real)),ring_1_of_int_real(M))) ) ).

% int_less_real_le
tff(fact_1292_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(divide_divide_real(ring_1_of_int_real(X)),ring_1_of_int_real(D)) = aa_real_real(plus_plus_real(ring_1_of_int_real(divide_divide_int(X,D))),aa_real_real(divide_divide_real(ring_1_of_int_real(modulo_modulo_int(X,D))),ring_1_of_int_real(D))) ).

% real_of_int_div_aux
tff(fact_1293_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(ring_1_of_int_real(N)),ring_1_of_int_real(X))),ring_1_of_int_real(divide_divide_int(N,X))))) ).

% real_of_int_div2
tff(fact_1294_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(ring_1_of_int_real(N)),ring_1_of_int_real(X))),ring_1_of_int_real(divide_divide_int(N,X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_1295_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ah(L)),Qr) ).

% divmod_step_nat_def
tff(fact_1296_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ai(L),Qr) ).

% divmod_step_int_def
tff(fact_1297_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_aj),divmod_nat(aa_nat_nat(minus_minus_nat(M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_1298_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_1299_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_1300_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_1301_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_1302_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_1303_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_1304_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_1305_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_1306_mask__half__int,axiom,
    ! [N: nat] : 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(minus_minus_nat(N),one_one_nat)) ).

% mask_half_int
tff(fact_1307_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(P,N3))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N3))) ) ) ) ).

% exists_least_lemma
tff(fact_1308_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = 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_1309_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(divide_divide_nat(M),N)),aa_nat_nat(modulo_modulo_nat(M),N)) ).

% divmod_nat_def
tff(fact_1310_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ak,N))) ).

% mask_eq_sum_exp_nat
tff(fact_1311_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_al),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_1312_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_am(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_1313_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_al),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(plus_plus_nat(N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(minus_minus_nat(M),one_one_nat)))),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_1314_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) ) )
       => ( 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 )
           => ( bit_se1409905431419307370or_int(K,L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( bit_se1409905431419307370or_int(K,L) = K ) )
              & ( ( L != zero_zero_int )
               => ( bit_se1409905431419307370or_int(K,L) = plus_plus_int(ord_max_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),modulo_modulo_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se1409905431419307370or_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_1315_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = numeral_numeral_nat(V) ) ) ).

% int_eq_iff_numeral
tff(fact_1316_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_1317_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_1318_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),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_1319_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_1320_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)),numeral_numeral_real(W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),numeral_numeral_nat(W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_1321_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_1322_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1323_or__minus__numerals_I6_J,axiom,
    ! [N: num] : 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_1324_or__minus__numerals_I2_J,axiom,
    ! [N: num] : 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_1325_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),bit_se1409905431419307370or_int(K,L))) ) ).

% or_greater_eq
tff(fact_1326_OR__lower,axiom,
    ! [X: int,Y2: 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),Y2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),bit_se1409905431419307370or_int(X,Y2))) ) ) ).

% OR_lower
tff(fact_1327_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,numeral_numeral_nat(N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_1328_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_1329_int__cases,axiom,
    ! [Z: int] :
      ( ! [N3: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% int_cases
tff(fact_1330_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z: int] :
      ( ! [N3: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N3)))
     => ( ! [N3: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3)))))
       => pp(aa_int_bool(P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_1331_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_1332_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_1333_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))
     => ? [N3: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% zero_le_imp_eq_int
tff(fact_1334_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N3: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% nonneg_int_cases
tff(fact_1335_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(plus_plus_nat(A),B)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_1336_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(plus_plus_nat(N),M)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_1337_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M),plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N),Z)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(plus_plus_nat(M),N)),Z) ).

% zadd_int_left
tff(fact_1338_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)) = times_times_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_1339_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_1340_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 = plus_plus_int(W,aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_1341_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(A),B)) = divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_1342_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_1343_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_1344_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y: real] :
        ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),X))) ) ).

% reals_Archimedean3
tff(fact_1345_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ) ).

% int_cases4
tff(fact_1346_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(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_1347_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_1348_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N),one_one_int) ).

% int_Suc
tff(fact_1349_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A),one_one_int) ).

% int_ops(4)
tff(fact_1350_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 = plus_plus_int(W,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_1351_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N3: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% nonpos_int_cases
tff(fact_1352_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_1353_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(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_1354_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% pos_int_cases
tff(fact_1355_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1356_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) )
       => ~ ! [N3: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ) ).

% int_cases3
tff(fact_1357_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(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_1358_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(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_1359_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),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,times_times_int(aa_nat_int(semiri1314217659103216013at_int,K),I2)),times_times_int(aa_nat_int(semiri1314217659103216013at_int,K),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1360_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_1361_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N3: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% negD
tff(fact_1362_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_1363_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(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(X),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(modulo_modulo_nat(X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_1364_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))
       => ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M5)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1365_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N3: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% neg_int_cases
tff(fact_1366_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y2: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(minus_minus_nat(X),Y2))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),X))
         => pp(aa_int_bool(P,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X),aa_nat_int(semiri1314217659103216013at_int,Y2)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_1367_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(divide_divide_nat(N),X))))) ).

% real_of_nat_div2
tff(fact_1368_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(divide_divide_nat(N),X)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_1369_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(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(X),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_1370_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(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_1371_OR__upper,axiom,
    ! [X: int,N: nat,Y2: 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,Y2),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,bit_se1409905431419307370or_int(X,Y2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_1372_or__int__rec,axiom,
    ! [K: int,L: int] : bit_se1409905431419307370or_int(K,L) = 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)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se1409905431419307370or_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_1373_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_1374_or__minus__numerals_I1_J,axiom,
    ! [N: num] : 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,bit_or_not_num_neg(one,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_1375_or__minus__numerals_I5_J,axiom,
    ! [N: num] : 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,bit_or_not_num_neg(one,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_1376_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se1412395901928357646or_nat(numeral_numeral_nat(aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_1377_or__nat__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit1,Y2))) = numeral_numeral_nat(aa_num_num(bit1,Y2)) ).

% or_nat_numerals(2)
tff(fact_1378_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se1412395901928357646or_nat(numeral_numeral_nat(aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_1379_or__nat__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit0,Y2))) = numeral_numeral_nat(aa_num_num(bit1,Y2)) ).

% or_nat_numerals(1)
tff(fact_1380_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : 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,bit_or_not_num_neg(M,aa_num_num(bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_1381_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : 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,bit_or_not_num_neg(M,aa_num_num(bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_1382_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : 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,bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_1383_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : 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,bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_1384_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one,one) = one ).

% or_not_num_neg.simps(1)
tff(fact_1385_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa_num_num(bit0,N),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_1386_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa_num_num(bit0,N),aa_num_num(bit1,M)) = aa_num_num(bit0,bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(6)
tff(fact_1387_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa_num_num(bit1,N),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_1388_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one,aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_1389_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa_num_num(bit0,N),aa_num_num(bit0,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(5)
tff(fact_1390_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa_num_num(bit1,N),aa_num_num(bit1,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(9)
tff(fact_1391_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one,aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_1392_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa_num_num(bit1,N),aa_num_num(bit0,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(8)
tff(fact_1393_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y2: num] :
      ( ( bit_or_not_num_neg(X,Xa2) = Y2 )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y2 != one ) ) )
       => ( ( ( X = one )
           => ! [M5: num] :
                ( ( Xa2 = aa_num_num(bit0,M5) )
               => ( Y2 != aa_num_num(bit1,M5) ) ) )
         => ( ( ( X = one )
             => ! [M5: num] :
                  ( ( Xa2 = aa_num_num(bit1,M5) )
                 => ( Y2 != aa_num_num(bit1,M5) ) ) )
           => ( ( ? [N3: num] : X = aa_num_num(bit0,N3)
               => ( ( Xa2 = one )
                 => ( Y2 != aa_num_num(bit0,one) ) ) )
             => ( ! [N3: num] :
                    ( ( X = aa_num_num(bit0,N3) )
                   => ! [M5: num] :
                        ( ( Xa2 = aa_num_num(bit0,M5) )
                       => ( Y2 != bitM(bit_or_not_num_neg(N3,M5)) ) ) )
               => ( ! [N3: num] :
                      ( ( X = aa_num_num(bit0,N3) )
                     => ! [M5: num] :
                          ( ( Xa2 = aa_num_num(bit1,M5) )
                         => ( Y2 != aa_num_num(bit0,bit_or_not_num_neg(N3,M5)) ) ) )
                 => ( ( ? [N3: num] : X = aa_num_num(bit1,N3)
                     => ( ( Xa2 = one )
                       => ( Y2 != one ) ) )
                   => ( ! [N3: num] :
                          ( ( X = aa_num_num(bit1,N3) )
                         => ! [M5: num] :
                              ( ( Xa2 = aa_num_num(bit0,M5) )
                             => ( Y2 != bitM(bit_or_not_num_neg(N3,M5)) ) ) )
                     => ~ ! [N3: num] :
                            ( ( X = aa_num_num(bit1,N3) )
                           => ! [M5: num] :
                                ( ( Xa2 = aa_num_num(bit1,M5) )
                               => ( Y2 != bitM(bit_or_not_num_neg(N3,M5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_1394_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(plus_plus_nat(N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_1395_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(plus_plus_nat(N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_1396_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(M),N) = aa_nat_nat(plus_plus_nat(zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_1397_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(bit_se1412395901928357646or_nat(M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(bit_se1412395901928357646or_nat(M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(bit_se1412395901928357646or_nat(M),N) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_1398_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),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(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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_1399_ln__le__cancel__iff,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1400_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_1401_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_1402_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_1403_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_1404_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_1405_ln__bound,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)),X)) ) ).

% ln_bound
tff(fact_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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(plus_plus_real(one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_1412_ln__mult,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y2)) ) ) ) ).

% ln_mult
tff(fact_1413_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_1414_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_1415_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_1416_ln__diff__le,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y2))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),Y2)),Y2))) ) ) ).

% ln_diff_le
tff(fact_1417_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(plus_plus_real(one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_1418_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_1419_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_1420_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_an(G)),X)) ) ).

% sums_if'
tff(fact_1421_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y2: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => ( pp(aa_real_bool(sums_real(F),Y2))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ao(G),F)),aa_real_real(plus_plus_real(X),Y2))) ) ) ).

% sums_if
tff(fact_1422_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ap(F),G),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aTP_Lamm_aq(F),set_ord_lessThan_nat(N))),groups6591440286371151544t_real(aTP_Lamm_ar(G),set_ord_lessThan_nat(N))) ).

% sum_split_even_odd
tff(fact_1423_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_1424_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_as),one_one_real)) ).

% power_half_series
tff(fact_1425_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I2,aa_nat_nat(plus_plus_nat(I2),J))) = groups1705073143266064639nt_int(aTP_Lamm_ag,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(plus_plus_nat(I2),J)))) ).

% prod_int_plus_eq
tff(fact_1426_sumr__cos__zero__one,axiom,
    ! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_at,set_ord_lessThan_nat(aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_1427_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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_au,X)) ) ) ) ).

% ln_series
tff(fact_1428_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),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(plus_plus_real(one_one_real),X))),X))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_1429_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_1430_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),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(plus_plus_real(one_one_real),X))),X))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_1431_tanh__real__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ).

% tanh_real_le_iff
tff(fact_1432_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_1433_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_1434_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_1435_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_1436_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_1437_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_1438_sin__bound__lemma,axiom,
    ! [X: real,Y2: real,U: real,V: real] :
      ( ( X = Y2 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
       => 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(plus_plus_real(X),U)),Y2))),V)) ) ) ).

% sin_bound_lemma
tff(fact_1439_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_1440_lemma__interval,axiom,
    ! [A: real,X: real,B: 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))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [Y: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y))),D2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_1441_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(plus_plus_real(one_one_real),X))),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_1442_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(aa_real_fun_nat_real(aTP_Lamm_av,X)) ) ).

% monoseq_arctan_series
tff(fact_1443_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(aa_real_fun_nat_real(aTP_Lamm_aw,X)) ) ) ).

% arctan_series
tff(fact_1444_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ax) ).

% pi_series
tff(fact_1445_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(aa_real_fun_nat_real(aTP_Lamm_aw,X)) ) ).

% summable_arctan_series
tff(fact_1446_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_1447_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_1448_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_1449_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_arctan_iff
tff(fact_1450_arctan__monotone_H,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y2))) ) ).

% arctan_monotone'
tff(fact_1451_arctan__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ).

% arctan_le_iff
tff(fact_1452_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_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_1453_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_1454_arctan__ubound,axiom,
    ! [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_1455_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_1456_arctan__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y2)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_1457_arctan__lbound,axiom,
    ! [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y2))) ).

% arctan_lbound
tff(fact_1458_dvd__imp__le__int,axiom,
    ! [I2: int,D: int] :
      ( ( I2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I2))
       => 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,I2))) ) ) ).

% dvd_imp_le_int
tff(fact_1459_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,modulo_modulo_int(K,L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_1460_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N3))),aa_nat_real(G,N3))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ay(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_1461_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ay(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_ay(F)))) ) ).

% summable_rabs
tff(fact_1462_machin__Euler,axiom,
    aa_real_real(plus_plus_real(aa_real_real(times_times_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),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_1463_machin,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_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),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),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_1464_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_1465_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) ).

% pi_ge_two
tff(fact_1466_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,times_times_int(M,N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_1467_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_1468_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))),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))),plus_plus_int(K,L))) ) ).

% even_abs_add_iff
tff(fact_1469_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))),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))),plus_plus_int(K,L))) ) ).

% even_add_abs_iff
tff(fact_1470_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_1471_pi__half__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_1472_pi__half__le__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_le_two
tff(fact_1473_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(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% monoseq_realpow
tff(fact_1474_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real))
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
       => ( 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_az(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_1475_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I3)),aa_nat_int(F,I3)))),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)))
           => ? [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1476_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),plus_plus_int(X,times_times_int(plus_plus_int(aa_int_int(abs_abs_int,minus_minus_int(X,Z)),one_one_int),D)))) ) ).

% incr_lemma
tff(fact_1477_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,minus_minus_int(X,times_times_int(plus_plus_int(aa_int_int(abs_abs_int,minus_minus_int(X,Z)),one_one_int),D))),Z)) ) ).

% decr_lemma
tff(fact_1478_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_1479_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% pi_half_ge_zero
tff(fact_1480_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_1481_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I3)),aa_nat_int(F,I3)))),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)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1482_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,real_V1022390504157884413omplex(X))),real_V1022390504157884413omplex(X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_1483_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),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_1484_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(real_V1022390504157884413omplex(plus_plus_complex(B,A))),real_V1022390504157884413omplex(B))),real_V1022390504157884413omplex(A))) ).

% complex_mod_triangle_ineq2
tff(fact_1485_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(F,aa_nat_nat(plus_plus_nat(I3),one_one_nat)),aa_nat_int(F,I3)))),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)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1486_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_ba,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_bb(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_1487_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_ba,collect_complex(aTP_Lamm_bc(N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_1488_arctan__add,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real))
       => ( aa_real_real(plus_plus_real(aa_real_real(arctan,X)),aa_real_real(arctan,Y2)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y2)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(X),Y2)))) ) ) ) ).

% arctan_add
tff(fact_1489_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(F,aa_nat_nat(plus_plus_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(plus_plus_nat(K),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_1490_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(times_times_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(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),X)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_1491_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_1492_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_1493_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_1494_cos__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi2
tff(fact_1495_cos__periodic__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi
tff(fact_1496_sin__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi2
tff(fact_1497_sin__periodic__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi
tff(fact_1498_sin__npi2,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).

% sin_npi2
tff(fact_1499_sin__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_1500_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),ring_1_of_int_real(N))) = zero_zero_real ).

% sin_npi_int
tff(fact_1501_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_1502_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_1503_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_1504_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_1505_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_1506_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_1507_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_1508_cos__npi2,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi2
tff(fact_1509_cos__npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi
tff(fact_1510_sin__2npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_1511_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_1512_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_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_1513_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),ring_1_of_int_real(N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_1514_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),ring_1_of_int_real(N))) = one_one_real ).

% cos_int_2pin
tff(fact_1515_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_1516_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_1517_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(times_times_real(pi),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(times_times_real(pi),ring_1_of_int_real(N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% cos_npi_int
tff(fact_1518_polar__Ex,axiom,
    ! [X: real,Y2: real] :
    ? [R3: real,A5: real] :
      ( ( X = aa_real_real(times_times_real(R3),aa_real_real(cos_real,A5)) )
      & ( Y2 = aa_real_real(times_times_real(R3),aa_real_real(sin_real,A5)) ) ) ).

% polar_Ex
tff(fact_1519_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_1520_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_1521_sin__x__le__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(sin_real,X)),X)) ) ).

% sin_x_le_x
tff(fact_1522_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_1523_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_1524_abs__sin__x__le__abs__x,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))),aa_real_real(abs_abs_real,X))) ).

% abs_sin_x_le_abs_x
tff(fact_1525_sin__cos__le1,axiom,
    ! [X: real,Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))))),one_one_real)) ).

% sin_cos_le1
tff(fact_1526_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_1527_sin__ge__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_eq_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_ge_zero
tff(fact_1528_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_1529_cos__monotone__0__pi__le,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_1530_cos__mono__le__eq,axiom,
    ! [X: real,Y2: 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),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_1531_cos__inj__pi,axiom,
    ! [X: real,Y2: 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),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),pi))
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y2) )
             => ( X = Y2 ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_1532_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_1533_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_1534_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_1535_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_1536_cos__monotone__0__pi,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_1537_cos__mono__less__eq,axiom,
    ! [X: real,Y2: 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),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_1538_cos__monotone__minus__pi__0_H,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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,Y2)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_1539_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] : X = aa_real_real(times_times_real(ring_1_of_int_real(I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_1540_sincos__total__pi,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
     => ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),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) )
            & ( Y2 = aa_real_real(sin_real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_1541_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_1542_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_1543_sin__gt__zero__02,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),numeral_numeral_real(aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_1544_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_1545_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_1546_cos__is__zero,axiom,
    ? [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),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X3) = zero_zero_real )
      & ! [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),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y) = zero_zero_real ) )
         => ( Y = X3 ) ) ) ).

% cos_is_zero
tff(fact_1547_cos__monotone__minus__pi__0,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),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,Y2)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_1548_cos__total,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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) = Y2 )
            & ! [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),pi))
                  & ( aa_real_real(cos_real,Y) = Y2 ) )
               => ( Y = X3 ) ) ) ) ) ).

% cos_total
tff(fact_1549_sincos__total__pi__half,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),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),numeral_numeral_real(aa_num_num(bit0,one)))))
              & ( X = aa_real_real(cos_real,T2) )
              & ( Y2 = aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_1550_sincos__total__2pi__le,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),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(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))
          & ( X = aa_real_real(cos_real,T2) )
          & ( Y2 = aa_real_real(sin_real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_1551_sincos__total__2pi,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),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(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T2) )
               => ( Y2 != aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_1552_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_1553_sin__gt__zero2,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_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero2
tff(fact_1554_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_1555_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),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(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_1556_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_30
tff(fact_1557_cos__gt__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),aa_real_real(divide_divide_real(pi),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
tff(fact_1558_sin__inj__pi,axiom,
    ! [X: real,Y2: 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),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),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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y2) )
             => ( X = Y2 ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_1559_sin__mono__le__eq,axiom,
    ! [X: real,Y2: 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),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),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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),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,Y2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_1560_sin__monotone__2pi__le,axiom,
    ! [Y2: 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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y2)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_1561_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_60
tff(fact_1562_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X4: int] : X = aa_real_real(times_times_real(aa_real_real(times_times_real(ring_1_of_int_real(X4)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_1563_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_1564_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)),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_1565_sin__mono__less__eq,axiom,
    ! [X: real,Y2: 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),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),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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),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,Y2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_1566_sin__monotone__2pi,axiom,
    ! [Y2: 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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y2)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_1567_sin__total,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
            & ( aa_real_real(sin_real,X3) = Y2 )
            & ! [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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
                  & ( aa_real_real(sin_real,Y) = Y2 ) )
               => ( Y = X3 ) ) ) ) ) ).

% sin_total
tff(fact_1568_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),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),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_1569_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),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),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_1570_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X4: nat] : X = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
        | ? [X4: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_1571_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_gt_0
tff(fact_1572_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X = aa_real_real(times_times_real(ring_1_of_int_real(I4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_1573_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X = aa_real_real(times_times_real(ring_1_of_int_real(I4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_1574_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_1575_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,numeral_numeral_nat(aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_1576_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_1577_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,numeral_numeral_nat(aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_1578_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bd,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_1579_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bd,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_1580_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bd,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_1581_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_be,X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_1582_tan__periodic__pi,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),pi)) = aa_real_real(tan_real,X) ).

% tan_periodic_pi
tff(fact_1583_tan__npi,axiom,
    ! [N: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_1584_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(numeral_numeral_real(N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_1585_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_1586_tan__periodic__int,axiom,
    ! [X: real,I2: int] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(ring_1_of_int_real(I2)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_1587_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_1588_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) ).

% square_fact_le_2_fact
tff(fact_1589_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_1590_lemma__tan__total,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y2))
     => ? [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(tan_real,X3))) ) ) ).

% lemma_tan_total
tff(fact_1591_tan__gt__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),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_gt_zero
tff(fact_1592_lemma__tan__total1,axiom,
    ! [Y2: 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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X3) = Y2 ) ) ).

% lemma_tan_total1
tff(fact_1593_tan__mono__lt__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),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),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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),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,Y2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_1594_tan__monotone_H,axiom,
    ! [Y2: 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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y2)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_1595_tan__monotone,axiom,
    ! [Y2: 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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y2)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_1596_tan__total,axiom,
    ! [Y2: 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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X3) = Y2 )
      & ! [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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y) = Y2 ) )
         => ( Y = X3 ) ) ) ).

% tan_total
tff(fact_1597_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_1598_tan__inverse,axiom,
    ! [Y2: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y2)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y2)) ).

% tan_inverse
tff(fact_1599_tan__pos__pi2__le,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),aa_real_real(divide_divide_real(pi),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(tan_real,X))) ) ) ).

% tan_pos_pi2_le
tff(fact_1600_tan__total__pos,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
     => ? [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_real,X3),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X3) = Y2 ) ) ) ).

% tan_total_pos
tff(fact_1601_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)),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_1602_tan__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),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),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),numeral_numeral_real(aa_num_num(bit0,one))))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),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,Y2)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_1603_tan__mono__le,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),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,Y2))) ) ) ) ).

% tan_mono_le
tff(fact_1604_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),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_1605_arctan__unique,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X) = Y2 )
         => ( aa_real_real(arctan,Y2) = X ) ) ) ) ).

% arctan_unique
tff(fact_1606_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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_1607_arctan,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y2)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y2)) = Y2 ) ) ).

% arctan
tff(fact_1608_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B6: real] : aa_real_real(F,H) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_bf(H),J),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(B6),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_1609_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))
     => ? [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z3) = X ) ) ) ).

% tan_total_pi4
tff(fact_1610_cos__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(cos_coeff,X2) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(divide_divide_nat(X2),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(cos_coeff,X2) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_1611_cos__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_bg,X)),aa_real_real(cos_real,X))) ).

% cos_paired
tff(fact_1612_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bh,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_1613_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bh,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_1614_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bh,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_1615_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] : aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bh,X),set_ord_lessThan_nat(N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_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(aa_real_fun_nat_real(power_power_real,X),N))) ).

% Maclaurin_sin_expansion
tff(fact_1616_sin__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(sin_coeff,X2) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(sin_coeff,X2) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(X2),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2)) ) ) ) ).

% sin_coeff_def
tff(fact_1617_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_1618_fact__mono__nat,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(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_1619_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_1620_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_1621_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_1622_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(minus_minus_nat(aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(M),N))) ) ) ).

% fact_diff_Suc
tff(fact_1623_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(divide_divide_nat(aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(N),R2)))),aa_nat_nat(power_power_nat(N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_1624_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)),groups708209901874060359at_nat(aTP_Lamm_al,set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_1625_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(divide_divide_nat(aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_al,set_or1269000886237332187st_nat(aa_nat_nat(plus_plus_nat(N),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_1626_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_1627_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_1628_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),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(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_1629_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),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(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_1630_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( 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(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))
             => ( Z != complex2(aa_real_real(cos_real,T2),aa_real_real(sin_real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_1631_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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bi,X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_1632_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_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,numeral_numeral_nat(aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(binomial(N),K) = aa_nat_nat(binomial(N),aa_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,numeral_numeral_nat(aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(binomial(N),K) = aa_nat_nat(divide_divide_nat(aa_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(N),K)),one_one_nat),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_1633_real__sqrt__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(sqrt,X) = aa_real_real(sqrt,Y2) )
    <=> ( X = Y2 ) ) ).

% real_sqrt_eq_iff
tff(fact_1634_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_1635_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_1636_real__sqrt__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ).

% real_sqrt_less_iff
tff(fact_1637_real__sqrt__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ).

% real_sqrt_le_iff
tff(fact_1638_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(binomial(aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_1639_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_1640_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_1641_exp__le__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ).

% exp_le_cancel_iff
tff(fact_1642_binomial__n__n,axiom,
    ! [N: nat] : aa_nat_nat(binomial(N),N) = one_one_nat ).

% binomial_n_n
tff(fact_1643_real__sqrt__gt__0__iff,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y2)) ) ).

% real_sqrt_gt_0_iff
tff(fact_1644_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_1645_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(binomial(zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_1646_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(binomial(N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_1647_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_1648_real__sqrt__ge__0__iff,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2)) ) ).

% real_sqrt_ge_0_iff
tff(fact_1649_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_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_1650_real__sqrt__gt__1__iff,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y2)) ) ).

% real_sqrt_gt_1_iff
tff(fact_1651_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_1652_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(binomial(aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(binomial(N),K)),aa_nat_nat(binomial(N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_1653_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_1654_real__sqrt__ge__1__iff,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y2)) ) ).

% real_sqrt_ge_1_iff
tff(fact_1655_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(binomial(N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_1656_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_1657_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_1658_real__sqrt__abs2,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(times_times_real(X),X)) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs2
tff(fact_1659_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_1660_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(binomial(N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_1661_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_1662_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_1663_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_1664_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_1665_norm__cos__sin,axiom,
    ! [T: real] : real_V1022390504157884413omplex(complex2(aa_real_real(cos_real,T),aa_real_real(sin_real,T))) = one_one_real ).

% norm_cos_sin
tff(fact_1666_real__sqrt__abs,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs
tff(fact_1667_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),numeral_numeral_nat(aa_num_num(bit0,one))) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_1668_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),numeral_numeral_nat(aa_num_num(bit0,one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_1669_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y2: real,Xa2: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(aa_num_num(bit0,one))))))),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_1670_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),K) ).

% real_sqrt_power
tff(fact_1671_choose__one,axiom,
    ! [N: nat] : aa_nat_nat(binomial(N),one_one_nat) = N ).

% choose_one
tff(fact_1672_real__sqrt__le__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2))) ) ).

% real_sqrt_le_mono
tff(fact_1673_real__sqrt__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2))) ) ).

% real_sqrt_less_mono
tff(fact_1674_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_1675_real__sqrt__mult,axiom,
    ! [X: real,Y2: real] : aa_real_real(sqrt,aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)) ).

% real_sqrt_mult
tff(fact_1676_real__sqrt__divide,axiom,
    ! [X: real,Y2: real] : aa_real_real(sqrt,aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)) ).

% real_sqrt_divide
tff(fact_1677_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(binomial(N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_1678_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(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(binomial(N),K)) ).

% Suc_times_binomial
tff(fact_1679_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(binomial(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(suc,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_1680_real__sqrt__gt__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,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_1681_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(binomial(N),K) = aa_nat_nat(binomial(N),aa_nat_nat(minus_minus_nat(N),K)) ) ) ).

% binomial_symmetric
tff(fact_1682_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_1683_real__sqrt__ge__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_eq_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_1684_exp__ge__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_ge_zero
tff(fact_1685_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_1686_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(M),R2)),K)),aa_nat_nat(plus_plus_nat(M),K))),aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(M),R2)),K)),K)),aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(M),R2)),M)) ).

% choose_mult_lemma
tff(fact_1687_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_1688_binomial__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(binomial(N),R2)),aa_nat_nat(power_power_nat(N),R2))) ) ).

% binomial_le_pow
tff(fact_1689_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( complex2(A,B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = numeral_numeral_real(W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_1690_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] : plus_plus_complex(complex2(A,B),complex2(C,D)) = complex2(aa_real_real(plus_plus_real(A),C),aa_real_real(plus_plus_real(B),D)) ).

% complex_add
tff(fact_1691_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(binomial(N),K))) ) ).

% zero_less_binomial
tff(fact_1692_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(binomial(aa_nat_nat(suc,aa_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(binomial(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_1693_complex__norm,axiom,
    ! [X: real,Y2: real] : real_V1022390504157884413omplex(complex2(X,Y2)) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_1694_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(binomial(N),M)),aa_nat_nat(binomial(M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(N),K)),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),K)),aa_nat_nat(minus_minus_nat(M),K))) ) ) ) ).

% choose_mult
tff(fact_1695_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(minus_minus_nat(N),K)))),aa_nat_nat(binomial(N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_1696_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(binomial(aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(binomial(N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_1697_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_1698_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(divide_divide_real(X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_1699_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(N),K)),aa_nat_nat(binomial(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_1700_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(X),Y2))),aa_real_real(plus_plus_real(aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_1701_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(exp_real,X))) ).

% exp_ge_add_one_self
tff(fact_1702_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2))))) ).

% le_real_sqrt_sumsq
tff(fact_1703_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( complex2(A,B) = uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_1704_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(complex2(A,B)),complex2(C,D)) = complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C))) ).

% complex_mult
tff(fact_1705_one__complex_Ocode,axiom,
    one_one_complex = complex2(one_one_real,zero_zero_real) ).

% one_complex.code
tff(fact_1706_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( complex2(A,B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_1707_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_1708_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(binomial(N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_1709_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(binomial(N),K) = aa_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(minus_minus_nat(N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_1710_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(plus_plus_real(one_one_real),X)),aa_real_real(exp_real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_1711_lemma__exp__total,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y2))
     => ? [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(Y2),one_one_real)))
          & ( aa_real_real(exp_real,X3) = Y2 ) ) ) ).

% lemma_exp_total
tff(fact_1712_ln__ge__iff,axiom,
    ! [X: real,Y2: 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,Y2),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y2)),X)) ) ) ).

% ln_ge_iff
tff(fact_1713_ln__x__over__x__mono,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Y2)),Y2)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_1714_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( complex2(A,B) = uminus1482373934393186551omplex(one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_1715_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),K)),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_1716_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K4)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(binomial(N),K)),aa_nat_nat(binomial(N),K4))) ) ) ).

% binomial_mono
tff(fact_1717_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(binomial(N),K4)),aa_nat_nat(binomial(N),K))) ) ) ) ).

% binomial_antimono
tff(fact_1718_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(binomial(N),K)),aa_nat_nat(binomial(N),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_1719_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(binomial(N),K)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_1720_real__less__rsqrt,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),Y2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y2))) ) ).

% real_less_rsqrt
tff(fact_1721_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(binomial(N),K) = aa_nat_nat(plus_plus_nat(aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),aa_nat_nat(minus_minus_nat(K),one_one_nat))),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_1722_real__le__rsqrt,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),Y2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,Y2))) ) ).

% real_le_rsqrt
tff(fact_1723_sqrt__le__D,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% sqrt_le_D
tff(fact_1724_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(binomial(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),aa_nat_nat(minus_minus_nat(K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_1725_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_1726_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(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(binomial(N),K)),aa_nat_nat(binomial(N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_1727_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K4))
     => ( 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,numeral_numeral_nat(aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(binomial(N),K4)),aa_nat_nat(binomial(N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_1728_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K4)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(binomial(N),K)),aa_nat_nat(binomial(N),K4))) ) ) ).

% binomial_strict_mono
tff(fact_1729_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(binomial(N),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))))) = aa_nat_nat(binomial(N),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_1730_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(binomial(N),aa_nat_nat(suc,K)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_1731_real__sqrt__unique,axiom,
    ! [Y2: real,X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))) = X )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
       => ( aa_real_real(sqrt,X) = Y2 ) ) ) ).

% real_sqrt_unique
tff(fact_1732_real__le__lsqrt,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y2)) ) ) ) ).

% real_le_lsqrt
tff(fact_1733_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_1734_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))) = Y2 )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_1735_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))) = X )
     => ( Y2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_1736_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),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% exp_half_le2
tff(fact_1737_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(A),C)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(B),D)),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,C),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),numeral_numeral_nat(aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_1738_real__sqrt__sum__squares__ge2,axiom,
    ! [Y2: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_1739_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_1740_sqrt__ge__absD,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),Y2)) ) ).

% sqrt_ge_absD
tff(fact_1741_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_45
tff(fact_1742_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_45
tff(fact_1743_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_1744_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(binomial(N),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(minus_minus_nat(N),one_one_nat))),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% choose_two
tff(fact_1745_real__less__lsqrt,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y2)) ) ) ) ).

% real_less_lsqrt
tff(fact_1746_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(X),Y2))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_1747_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_1748_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs1
tff(fact_1749_real__sqrt__ge__abs2,axiom,
    ! [Y2: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs2
tff(fact_1750_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y2)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_1751_ln__sqrt,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(sqrt,X)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_1752_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_30
tff(fact_1753_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_60
tff(fact_1754_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_1755_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(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_1756_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y2: real,Xa2: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_1757_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_1758_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(X),Y2))),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y2)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_1759_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),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,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_1760_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),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(plus_plus_real(one_one_real),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_1761_cos__x__y__le__one,axiom,
    ! [X: real,Y2: 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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))))),one_one_real)) ).

% cos_x_y_le_one
tff(fact_1762_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y2)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_1763_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(plus_plus_real(X),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_1764_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(aa_real_fun_nat_real(power_power_real,aa_real_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_1765_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(aa_real_fun_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_1766_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(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_1767_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(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_1768_Maclaurin__exp__le,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(exp_real,X) = aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bi,X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_1769_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(U),numeral_numeral_real(aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(divide_divide_real(U),numeral_numeral_real(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),Y2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_1770_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(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_1771_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(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_1772_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(X),aa_real_real(plus_plus_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_1773_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(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),X)))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_1774_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(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_1775_arsinh__real__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_1776_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_1777_sin__arccos__abs,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real))
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y2)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_1778_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_1779_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_1780_cos__arccos,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y2)) = Y2 ) ) ) ).

% cos_arccos
tff(fact_1781_sin__arcsin,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y2)) = Y2 ) ) ) ).

% sin_arcsin
tff(fact_1782_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_1783_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_1784_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),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_1785_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,K)) = set_ord_atMost_nat(K) ).

% lessThan_Suc_atMost
tff(fact_1786_arccos__le__arccos,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y2)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_1787_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bj(M)),set_ord_atMost_nat(N)) = aa_nat_nat(binomial(aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_1788_arccos__le__mono,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),X)) ) ) ) ).

% arccos_le_mono
tff(fact_1789_arccos__eq__iff,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real)) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y2) )
      <=> ( X = Y2 ) ) ) ).

% arccos_eq_iff
tff(fact_1790_arcsin__le__arcsin,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_1791_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_1792_arcsin__le__mono,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ) ).

% arcsin_le_mono
tff(fact_1793_arcsin__eq__iff,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real))
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% arcsin_eq_iff
tff(fact_1794_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bk(R2)),set_ord_atMost_nat(N)) = aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(R2),N))),N) ).

% sum_choose_lower
tff(fact_1795_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bl(N)),set_ord_atMost_nat(M)) = aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_1796_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bl(N)),set_ord_atMost_nat(M)) = aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(N),M)),one_one_nat)),aa_nat_nat(plus_plus_nat(N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_1797_arccos__lbound,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y2))) ) ) ).

% arccos_lbound
tff(fact_1798_arccos__less__arccos,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y2)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_1799_arccos__less__mono,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X)) ) ) ) ).

% arccos_less_mono
tff(fact_1800_arccos__ubound,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y2)),pi)) ) ) ).

% arccos_ubound
tff(fact_1801_arccos__cos,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),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_1802_arcsin__less__arcsin,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_1803_arcsin__less__mono,axiom,
    ! [X: real,Y2: 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,Y2)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ) ) ).

% arcsin_less_mono
tff(fact_1804_cos__arccos__abs,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real))
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y2)) = Y2 ) ) ).

% cos_arccos_abs
tff(fact_1805_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi))
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_1806_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_bm(M),N)),set_ord_atMost_nat(M)) = aa_nat_nat(binomial(aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_1807_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_bn(M),N),R2)),set_ord_atMost_nat(R2)) = aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(M),N)),R2) ).

% vandermonde
tff(fact_1808_arccos__lt__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y2)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_1809_arccos__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y2)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_1810_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_1811_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_1812_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_1813_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_1814_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(binomial(N)),set_ord_atMost_nat(N)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_1815_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(power_power_nat(aa_nat_nat(plus_plus_nat(A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_bo(A),B),N)),set_ord_atMost_nat(N)) ).

% binomial
tff(fact_1816_arccos,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y2)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y2)) = Y2 ) ) ) ) ).

% arccos
tff(fact_1817_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_1818_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_nat(A,I3) = 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_bp(A),X)),set_ord_atMost_nat(M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_bp(B),X)),set_ord_atMost_nat(N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_br(A),B),X)),set_ord_atMost_nat(aa_nat_nat(plus_plus_nat(M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_1819_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bs(N)),set_ord_atMost_nat(N)) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_1820_arccos__le__pi2,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_1821_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)),one_one_nat))),set_ord_atMost_nat(M)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_1822_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bt(N)),set_ord_atMost_nat(N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(N),one_one_nat))) ).

% choose_linear_sum
tff(fact_1823_arcsin__lt__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_1824_arcsin__lbound,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2))) ) ) ).

% arcsin_lbound
tff(fact_1825_arcsin__ubound,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).

% arcsin_ubound
tff(fact_1826_arcsin__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_bounded
tff(fact_1827_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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_1828_arcsin,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y2)) = Y2 ) ) ) ) ).

% arcsin
tff(fact_1829_arcsin__pi,axiom,
    ! [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y2)),pi))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y2)) = Y2 ) ) ) ) ).

% arcsin_pi
tff(fact_1830_arcsin__le__iff,axiom,
    ! [X: real,Y2: 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)),numeral_numeral_real(aa_num_num(bit0,one)))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),Y2))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sin_real,Y2))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_1831_le__arcsin__iff,axiom,
    ! [X: real,Y2: 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)),numeral_numeral_real(aa_num_num(bit0,one)))),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(arcsin,X)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y2)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_1832_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(times_times_real(ring_1_of_int_real(K2)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_1833_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_1834_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_1835_real__scaleR__def,axiom,
    ! [A: real,X: real] : aa_real_real(real_V1485227260804924795R_real(A),X) = aa_real_real(times_times_real(A),X) ).

% real_scaleR_def
tff(fact_1836_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : real_V2046097035970521341omplex(R2,complex2(A,B)) = complex2(aa_real_real(times_times_real(R2),A),aa_real_real(times_times_real(R2),B)) ).

% complex_scaleR
tff(fact_1837_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(aa_real_fun_nat_real(aTP_Lamm_bh,X),set_ord_lessThan_nat(N))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_1838_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)),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_1839_i__even__power,axiom,
    ! [N: nat] : power_power_complex(imaginary_unit,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),numeral_numeral_nat(aa_num_num(bit0,one)))) = power_power_complex(uminus1482373934393186551omplex(one_one_complex),N) ).

% i_even_power
tff(fact_1840_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,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_1841_log__one,axiom,
    ! [A: real] : aa_real_real(log(A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_1842_norm__ii,axiom,
    real_V1022390504157884413omplex(imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_1843_complex__i__mult__minus,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),X)) = uminus1482373934393186551omplex(X) ).

% complex_i_mult_minus
tff(fact_1844_divide__i,axiom,
    ! [X: complex] : divide1717551699836669952omplex(X,imaginary_unit) = aa_complex_complex(times_times_complex(uminus1482373934393186551omplex(imaginary_unit)),X) ).

% divide_i
tff(fact_1845_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(log(A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_1846_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y2: 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),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(A),X)),aa_real_real(log(A),Y2)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1847_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(log(A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1848_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(log(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1849_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(log(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_1850_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(log(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1851_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = uminus1482373934393186551omplex(one_one_complex) ).

% i_squared
tff(fact_1852_divide__numeral__i,axiom,
    ! [Z: complex,N: num] : divide1717551699836669952omplex(Z,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(N)),imaginary_unit)) = divide1717551699836669952omplex(uminus1482373934393186551omplex(aa_complex_complex(times_times_complex(imaginary_unit),Z)),numera6690914467698888265omplex(N)) ).

% divide_numeral_i
tff(fact_1853_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y2: 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),Y2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(A),X)),aa_real_real(log(A),Y2)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1854_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(log(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_1855_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(log(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1856_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(log(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_1857_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(log(A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1858_cot__npi,axiom,
    ! [N: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_1859_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(log(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_1860_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_1861_power2__i,axiom,
    power_power_complex(imaginary_unit,numeral_numeral_nat(aa_num_num(bit0,one))) = uminus1482373934393186551omplex(one_one_complex) ).

% power2_i
tff(fact_1862_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_1863_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_1864_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != numera6690914467698888265omplex(W) ).

% complex_i_not_numeral
tff(fact_1865_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(log(A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A),X)) ) ) ) ) ).

% log_inverse
tff(fact_1866_divide__real__def,axiom,
    ! [X: real,Y2: real] : aa_real_real(divide_divide_real(X),Y2) = aa_real_real(times_times_real(X),aa_real_real(inverse_inverse_real,Y2)) ).

% divide_real_def
tff(fact_1867_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
    <=> ( W = uminus1482373934393186551omplex(aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_1868_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_1869_log__ln,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = aa_real_real(log(aa_real_real(exp_real,one_one_real)),X) ).

% log_ln
tff(fact_1870_Complex__eq__i,axiom,
    ! [X: real,Y2: real] :
      ( ( complex2(X,Y2) = imaginary_unit )
    <=> ( ( X = zero_zero_real )
        & ( Y2 = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_1871_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero_real,one_one_real) ).

% imaginary_unit.code
tff(fact_1872_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(complex2(A,B)),imaginary_unit) = complex2(aa_real_real(uminus_uminus_real,B),A) ).

% Complex_mult_i
tff(fact_1873_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),complex2(A,B)) = complex2(aa_real_real(uminus_uminus_real,B),A) ).

% i_mult_Complex
tff(fact_1874_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(log(B),X) = aa_real_real(divide_divide_real(aa_real_real(log(A),X)),aa_real_real(log(A),B)) ) ) ) ).

% log_base_change
tff(fact_1875_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_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(log(B),M))) ) ) ).

% less_log_of_power
tff(fact_1876_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_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(log(B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_1877_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E2))
         => ( pp(aa_real_bool(P,D2))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [N3: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono_1
tff(fact_1878_real__arch__inverse,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
    <=> ? [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))),E)) ) ) ).

% real_arch_inverse
tff(fact_1879_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E2))
         => ( pp(aa_real_bool(P,D2))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono
tff(fact_1880_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_1881_log__mult,axiom,
    ! [A: real,X: real,Y2: 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),Y2))
           => ( aa_real_real(log(A),aa_real_real(times_times_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(log(A),X)),aa_real_real(log(A),Y2)) ) ) ) ) ) ).

% log_mult
tff(fact_1882_log__divide,axiom,
    ! [A: real,X: real,Y2: 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),Y2))
           => ( aa_real_real(log(A),aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(log(A),X)),aa_real_real(log(A),Y2)) ) ) ) ) ) ).

% log_divide
tff(fact_1883_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(aa_real_fun_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(log(B),M))) ) ) ).

% le_log_of_power
tff(fact_1884_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(log(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),X) = aa_real_real(divide_divide_real(aa_real_real(log(A),X)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_1885_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B),X)) ) ) ).

% log_nat_power
tff(fact_1886_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N) )
     => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_1887_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(aa_real_fun_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(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_1888_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(log(A),X) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(log(B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1889_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_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_1890_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(aa_real_fun_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(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_1891_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(plus_plus_real(X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_1892_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_1893_tan__cot,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),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_1894_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(power_power_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(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_1895_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_1896_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)))),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_1897_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)))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_1898_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(power_power_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(log(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_1899_cot__gt__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),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X))) ) ) ).

% cot_gt_zero
tff(fact_1900_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(power_power_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(log(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_1901_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(times_times_real(aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_1902_tan__cot_H,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_1903_Arg__minus__ii,axiom,
    arg(uminus1482373934393186551omplex(imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_1904_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,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))
       => ( ( archim7802044766580827645g_real(aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = 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(power_power_nat(B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(power_power_nat(B),aa_nat_nat(plus_plus_nat(N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_1905_Arg__ii,axiom,
    arg(imaginary_unit) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_1906_sinh__real__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ).

% sinh_real_le_iff
tff(fact_1907_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_1908_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_1909_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : archim7802044766580827645g_real(aa_real_real(divide_divide_real(numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_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_1910_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : archim7802044766580827645g_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_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_1911_divide__complex__def,axiom,
    ! [X: complex,Y2: complex] : divide1717551699836669952omplex(X,Y2) = aa_complex_complex(times_times_complex(X),invers8013647133539491842omplex(Y2)) ).

% divide_complex_def
tff(fact_1912_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),arg(Z)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,arg(Z)),pi)) ) ).

% Arg_bounded
tff(fact_1913_complex__inverse,axiom,
    ! [A: real,B: real] : invers8013647133539491842omplex(complex2(A,B)) = complex2(aa_real_real(divide_divide_real(A),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_1914_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))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_1915_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(power_power_nat(B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(power_power_nat(B),aa_nat_nat(plus_plus_nat(N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))
         => ( archim7802044766580827645g_real(aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_1916_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = plus_plus_int(archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(plus_plus_nat(aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(N),one_one_nat)),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat)))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_1917_cis__minus__pi__half,axiom,
    cis(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) = uminus1482373934393186551omplex(imaginary_unit) ).

% cis_minus_pi_half
tff(fact_1918_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))
       => ( ( archim7802044766580827645g_real(aa_real_real(log(B),X)) = 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(plus_plus_nat(K),one_one_nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_1919_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,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))
       => ( ( archim6058952711729229775r_real(aa_real_real(log(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(power_power_nat(B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(power_power_nat(B),aa_nat_nat(plus_plus_nat(N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_1920_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(A),X)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_1921_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_1922_norm__cis,axiom,
    ! [A: real] : real_V1022390504157884413omplex(cis(A)) = one_one_real ).

% norm_cis
tff(fact_1923_cis__zero,axiom,
    cis(zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_1924_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_1925_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_1926_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_1927_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_1928_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : aa_real_real(powr_real(numeral_numeral_real(M)),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),numeral_numeral_nat(N)) ).

% numeral_powr_numeral_real
tff(fact_1929_cis__pi,axiom,
    cis(pi) = uminus1482373934393186551omplex(one_one_complex) ).

% cis_pi
tff(fact_1930_log__powr__cancel,axiom,
    ! [A: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_real_real(powr_real(A),Y2)) = Y2 ) ) ) ).

% log_powr_cancel
tff(fact_1931_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(log(A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_1932_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : archim6058952711729229775r_real(aa_real_real(divide_divide_real(numeral_numeral_real(A)),numeral_numeral_real(B))) = divide_divide_int(aa_num_int(numeral_numeral_int,A),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_1933_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(powr_real(X),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(N)) ) ) ).

% powr_numeral
tff(fact_1934_cis__pi__half,axiom,
    cis(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_1935_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : archim6058952711729229775r_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(B))) = divide_divide_int(one_one_int,aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_1936_cis__2pi,axiom,
    cis(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_1937_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : archim6058952711729229775r_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(numeral_numeral_real(A)),numeral_numeral_real(B)))) = 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_1938_square__powr__half,axiom,
    ! [X: real] : aa_real_real(powr_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_1939_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : archim6058952711729229775r_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(B)))) = 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_1940_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_le_cosh_real
tff(fact_1941_powr__powr,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(X),A)),B) = aa_real_real(powr_real(X),aa_real_real(times_times_real(A),B)) ).

% powr_powr
tff(fact_1942_powr__ge__pzero,axiom,
    ! [X: real,Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(X),Y2))) ).

% powr_ge_pzero
tff(fact_1943_powr__mono2,axiom,
    ! [A: real,X: real,Y2: 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),Y2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A))) ) ) ) ).

% powr_mono2
tff(fact_1944_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_1945_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_1946_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_1947_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y2: 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,Y2),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_1948_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_1949_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_nonneg
tff(fact_1950_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_1951_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(cis(A)),cis(B)) = cis(aa_real_real(plus_plus_real(A),B)) ).

% cis_mult
tff(fact_1952_powr__less__mono2,axiom,
    ! [A: real,X: real,Y2: 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),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A))) ) ) ) ).

% powr_less_mono2
tff(fact_1953_powr__mono2_H,axiom,
    ! [A: real,X: real,Y2: 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,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(Y2),A)),aa_real_real(powr_real(X),A))) ) ) ) ).

% powr_mono2'
tff(fact_1954_powr__inj,axiom,
    ! [A: real,X: real,Y2: 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),Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% powr_inj
tff(fact_1955_gr__one__powr,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(powr_real(X),Y2))) ) ) ).

% gr_one_powr
tff(fact_1956_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_1957_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y2: 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),Y2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),B))) ) ) ) ) ).

% powr_mono_both
tff(fact_1958_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_1959_powr__divide,axiom,
    ! [X: real,Y2: real,A: 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),Y2))
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(X),Y2)),A) = aa_real_real(divide_divide_real(aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A)) ) ) ) ).

% powr_divide
tff(fact_1960_powr__mult,axiom,
    ! [X: real,Y2: real,A: 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),Y2))
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(X),Y2)),A) = aa_real_real(times_times_real(aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A)) ) ) ) ).

% powr_mult
tff(fact_1961_inverse__powr,axiom,
    ! [Y2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2))
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y2)),A) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y2),A)) ) ) ).

% inverse_powr
tff(fact_1962_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(powr_real(B),C)) = aa_real_real(times_times_real(A),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_1963_ln__powr,axiom,
    ! [X: real,Y2: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(X),Y2)) = aa_real_real(times_times_real(Y2),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_powr
tff(fact_1964_log__powr,axiom,
    ! [X: real,B: real,Y2: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(log(B),aa_real_real(powr_real(X),Y2)) = aa_real_real(times_times_real(Y2),aa_real_real(log(B),X)) ) ) ).

% log_powr
tff(fact_1965_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y2: 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,Y2),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_1966_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y2: 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),Y2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_1967_cosh__real__strict__mono,axiom,
    ! [X: real,Y2: 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),Y2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))) ) ) ).

% cosh_real_strict_mono
tff(fact_1968_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_1969_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))
       => ( ( archim6058952711729229775r_real(aa_real_real(log(B),X)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),ring_1_of_int_real(K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(powr_real(B),ring_1_of_int_real(plus_plus_int(K,one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_1970_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(powr_real(X),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N) ) ) ).

% powr_realpow
tff(fact_1971_less__log__iff,axiom,
    ! [B: real,X: real,Y2: 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,Y2),aa_real_real(log(B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),Y2)),X)) ) ) ) ).

% less_log_iff
tff(fact_1972_log__less__iff,axiom,
    ! [B: real,X: real,Y2: 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(log(B),X)),Y2))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(powr_real(B),Y2))) ) ) ) ).

% log_less_iff
tff(fact_1973_less__powr__iff,axiom,
    ! [B: real,X: real,Y2: 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),Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(B),X)),Y2)) ) ) ) ).

% less_powr_iff
tff(fact_1974_powr__less__iff,axiom,
    ! [B: real,X: real,Y2: 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),Y2)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),aa_real_real(log(B),X))) ) ) ) ).

% powr_less_iff
tff(fact_1975_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(plus_plus_real(ring_1_of_int_real(archim6058952711729229775r_real(R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_1976_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,ring_1_of_int_real(N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(ring_1_of_int_real(N)),one_one_real)))
       => ( archim6058952711729229775r_real(X) = N ) ) ) ).

% floor_eq
tff(fact_1977_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(plus_plus_real(ring_1_of_int_real(archim6058952711729229775r_real(R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_1978_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)),ring_1_of_int_real(archim6058952711729229775r_real(R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_1979_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)),ring_1_of_int_real(archim6058952711729229775r_real(R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_1980_DeMoivre,axiom,
    ! [A: real,N: nat] : power_power_complex(cis(A),N) = cis(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).

% DeMoivre
tff(fact_1981_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_1982_powr__mult__base,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(times_times_real(X),aa_real_real(powr_real(X),Y2)) = aa_real_real(powr_real(X),aa_real_real(plus_plus_real(one_one_real),Y2)) ) ) ).

% powr_mult_base
tff(fact_1983_powr__le__iff,axiom,
    ! [B: real,X: real,Y2: 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),Y2)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),aa_real_real(log(B),X))) ) ) ) ).

% powr_le_iff
tff(fact_1984_le__powr__iff,axiom,
    ! [B: real,X: real,Y2: 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),Y2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(B),X)),Y2)) ) ) ) ).

% le_powr_iff
tff(fact_1985_log__le__iff,axiom,
    ! [B: real,X: real,Y2: 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(log(B),X)),Y2))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(powr_real(B),Y2))) ) ) ) ).

% log_le_iff
tff(fact_1986_le__log__iff,axiom,
    ! [B: real,X: real,Y2: 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,Y2),aa_real_real(log(B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),Y2)),X)) ) ) ) ).

% le_log_iff
tff(fact_1987_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,ring_1_of_int_real(N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(ring_1_of_int_real(N)),one_one_real)))
       => ( archim6058952711729229775r_real(X) = N ) ) ) ).

% floor_eq2
tff(fact_1988_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))
     => ( archim6058952711729229775r_real(aa_real_real(divide_divide_real(A),ring_1_of_int_real(B))) = divide_divide_int(archim6058952711729229775r_real(A),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_1989_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_1990_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(times_times_real(aa_real_real(powr_real(A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_1991_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y2: 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(plus_plus_real(Y2),aa_real_real(log(B),X)) = aa_real_real(log(B),aa_real_real(times_times_real(aa_real_real(powr_real(B),Y2)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1992_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y2: 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(plus_plus_real(aa_real_real(log(B),X)),Y2) = aa_real_real(log(B),aa_real_real(times_times_real(X),aa_real_real(powr_real(B),Y2))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1993_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y2: 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(Y2),aa_real_real(log(B),X)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y2)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1994_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y2: 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(log(B),X)),Y2) = aa_real_real(log(B),aa_real_real(times_times_real(X),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y2)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_1995_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),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_1996_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,numeral_numeral_real(N))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(N))) ) ) ).

% powr_neg_numeral
tff(fact_1997_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(plus_plus_real(X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_1998_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( archim6058952711729229775r_real(aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = plus_plus_int(archim6058952711729229775r_real(aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_1999_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(power_power_nat(B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(power_power_nat(B),aa_nat_nat(plus_plus_nat(N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))
         => ( archim6058952711729229775r_real(aa_real_real(log(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_2000_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_bu(N),set_ord_lessThan_nat(N),collect_complex(aTP_Lamm_bc(N))) ) ).

% bij_betw_roots_unity
tff(fact_2001_exp__pi__i_H,axiom,
    exp_complex(aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(pi))) = uminus1482373934393186551omplex(one_one_complex) ).

% exp_pi_i'
tff(fact_2002_exp__pi__i,axiom,
    exp_complex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(pi)),imaginary_unit)) = uminus1482373934393186551omplex(one_one_complex) ).

% exp_pi_i
tff(fact_2003_exp__two__pi__i_H,axiom,
    exp_complex(aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_2004_exp__two__pi__i,axiom,
    exp_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),real_V4546457046886955230omplex(pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_2005_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A5: complex,R3: real] : Z = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R3)),exp_complex(A5)) ).

% complex_exp_exists
tff(fact_2006_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y2: real] : aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R2)),complex2(X,Y2)) = complex2(aa_real_real(times_times_real(R2),X),aa_real_real(times_times_real(R2),Y2)) ).

% complex_of_real_mult_Complex
tff(fact_2007_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y2: real,R2: real] : aa_complex_complex(times_times_complex(complex2(X,Y2)),real_V4546457046886955230omplex(R2)) = complex2(aa_real_real(times_times_real(X),R2),aa_real_real(times_times_real(Y2),R2)) ).

% Complex_mult_complex_of_real
tff(fact_2008_Complex__add__complex__of__real,axiom,
    ! [X: real,Y2: real,R2: real] : plus_plus_complex(complex2(X,Y2),real_V4546457046886955230omplex(R2)) = complex2(aa_real_real(plus_plus_real(X),R2),Y2) ).

% Complex_add_complex_of_real
tff(fact_2009_complex__of__real__add__Complex,axiom,
    ! [R2: real,X: real,Y2: real] : plus_plus_complex(real_V4546457046886955230omplex(R2),complex2(X,Y2)) = complex2(aa_real_real(plus_plus_real(R2),X),Y2) ).

% complex_of_real_add_Complex
tff(fact_2010_cis__conv__exp,axiom,
    ! [B: real] : cis(B) = exp_complex(aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(B))) ).

% cis_conv_exp
tff(fact_2011_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R2)),imaginary_unit) = complex2(zero_zero_real,R2) ).

% complex_of_real_i
tff(fact_2012_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(R2)) = complex2(zero_zero_real,R2) ).

% i_complex_of_real
tff(fact_2013_Complex__eq,axiom,
    ! [A: real,B: real] : complex2(A,B) = plus_plus_complex(real_V4546457046886955230omplex(A),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(B))) ).

% Complex_eq
tff(fact_2014_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A5: real] : Z = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R3)),plus_plus_complex(real_V4546457046886955230omplex(aa_real_real(cos_real,A5)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sin_real,A5))))) ).

% complex_split_polar
tff(fact_2015_cmod__unit__one,axiom,
    ! [A: real] : real_V1022390504157884413omplex(plus_plus_complex(real_V4546457046886955230omplex(aa_real_real(cos_real,A)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_2016_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R2)),plus_plus_complex(real_V4546457046886955230omplex(aa_real_real(cos_real,A)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_2017_csqrt__ii,axiom,
    csqrt(imaginary_unit) = divide1717551699836669952omplex(plus_plus_complex(one_one_complex,imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_2018_arctan__def,axiom,
    ! [Y2: real] : aa_real_real(arctan,Y2) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bv,Y2)) ).

% arctan_def
tff(fact_2019_arcsin__def,axiom,
    ! [Y2: real] : aa_real_real(arcsin,Y2) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bw,Y2)) ).

% arcsin_def
tff(fact_2020_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( sgn_sgn_int(L) = zero_zero_int )
          | ( sgn_sgn_int(K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( modulo_modulo_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,N))) = times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( sgn_sgn_int(L) = zero_zero_int )
            | ( sgn_sgn_int(K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( sgn_sgn_int(K) = sgn_sgn_int(L) )
           => ( modulo_modulo_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,N))) = times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(M),N))) ) )
          & ( ( sgn_sgn_int(K) != sgn_sgn_int(L) )
           => ( modulo_modulo_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,N))) = times_times_int(sgn_sgn_int(L),minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),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(modulo_modulo_nat(M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_2021_csqrt__1,axiom,
    csqrt(one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_2022_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_2023_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,times_times_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_2024_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,times_times_int(L,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_2025_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),times_times_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_2026_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),times_times_int(K,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_2027_power2__csqrt,axiom,
    ! [Z: complex] : power_power_complex(csqrt(Z),numeral_numeral_nat(aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_2028_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N3: nat,L2: int] : K != times_times_int(sgn_sgn_int(L2),aa_nat_int(semiri1314217659103216013at_int,N3)) ).

% int_sgnE
tff(fact_2029_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))
       => ( sgn_sgn_int(modulo_modulo_int(K,L)) = sgn_sgn_int(L) ) ) ) ).

% sgn_mod
tff(fact_2030_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_bx) ) ) ).

% ln_neg_is_const
tff(fact_2031_zsgn__def,axiom,
    ! [I2: int] :
      ( ( ( I2 = zero_zero_int )
       => ( sgn_sgn_int(I2) = zero_zero_int ) )
      & ( ( I2 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I2))
           => ( sgn_sgn_int(I2) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I2))
           => ( sgn_sgn_int(I2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_2032_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( divide_divide_int(times_times_int(sgn_sgn_int(V),aa_int_int(abs_abs_int,K)),times_times_int(sgn_sgn_int(V),aa_int_int(abs_abs_int,L))) = divide_divide_int(aa_int_int(abs_abs_int,K),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_2033_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
     => ( divide_divide_int(K,L) = times_times_int(times_times_int(sgn_sgn_int(K),sgn_sgn_int(L)),divide_divide_int(aa_int_int(abs_abs_int,K),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_2034_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( real_V4546457046886955230omplex(aa_real_real(sqrt,X)) = csqrt(real_V4546457046886955230omplex(X)) ) ) ).

% of_real_sqrt
tff(fact_2035_arccos__def,axiom,
    ! [Y2: real] : aa_real_real(arccos,Y2) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_by,Y2)) ).

% arccos_def
tff(fact_2036_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( sgn_sgn_int(R2) = 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 = plus_plus_int(times_times_int(Q2,L),R2) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q2,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2037_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 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != times_times_int(Q3,A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R3) )
               => ( ( sgn_sgn_int(R3) = sgn_sgn_int(A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != plus_plus_int(times_times_int(Q3,A22),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2038_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 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L3: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = times_times_int(Q4,L3) ) )
        | ? [R5: int,L3: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q4,R5) )
            & ( sgn_sgn_int(R5) = 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 = plus_plus_int(times_times_int(Q4,L3),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2039_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_bz) ).

% pi_half
tff(fact_2040_pi__def,axiom,
    pi = aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_bz)) ).

% pi_def
tff(fact_2041_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( sgn_sgn_int(L) = zero_zero_int )
          | ( sgn_sgn_int(K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( divide_divide_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( sgn_sgn_int(L) = zero_zero_int )
            | ( sgn_sgn_int(K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( sgn_sgn_int(K) = sgn_sgn_int(L) )
           => ( divide_divide_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(M),N)) ) )
          & ( ( sgn_sgn_int(K) != sgn_sgn_int(L) )
           => ( divide_divide_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_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(plus_plus_nat(aa_nat_nat(divide_divide_nat(M),N)),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_2042_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( modulo_modulo_int(K,L) = K ) )
      & ( ( L != zero_zero_int )
       => ( ( ( sgn_sgn_int(K) = sgn_sgn_int(L) )
           => ( modulo_modulo_int(K,L) = times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(nat2(aa_int_int(abs_abs_int,K))),nat2(aa_int_int(abs_abs_int,L))))) ) )
          & ( ( sgn_sgn_int(K) != sgn_sgn_int(L) )
           => ( modulo_modulo_int(K,L) = times_times_int(sgn_sgn_int(L),minus_minus_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(modulo_modulo_nat(nat2(aa_int_int(abs_abs_int,K))),nat2(aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_2043_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( divide_divide_int(K,L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( sgn_sgn_int(K) = sgn_sgn_int(L) )
           => ( divide_divide_int(K,L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(nat2(aa_int_int(abs_abs_int,K))),nat2(aa_int_int(abs_abs_int,L)))) ) )
          & ( ( sgn_sgn_int(K) != sgn_sgn_int(L) )
           => ( divide_divide_int(K,L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(plus_plus_nat(aa_nat_nat(divide_divide_nat(nat2(aa_int_int(abs_abs_int,K))),nat2(aa_int_int(abs_abs_int,L)))),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_2044_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa_num_num(bit1,X32)) = aa_nat_nat(plus_plus_nat(size_num(X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_2045_powr__int,axiom,
    ! [X: real,I2: 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),I2))
         => ( aa_real_real(powr_real(X),ring_1_of_int_real(I2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),nat2(I2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
         => ( aa_real_real(powr_real(X),ring_1_of_int_real(I2)) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),nat2(aa_int_int(uminus_uminus_int,I2)))) ) ) ) ) ).

% powr_int
tff(fact_2046_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_2047_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_sgn_iff
tff(fact_2048_nat__numeral,axiom,
    ! [K: num] : nat2(aa_num_int(numeral_numeral_int,K)) = numeral_numeral_nat(K) ).

% nat_numeral
tff(fact_2049_nat__1,axiom,
    nat2(one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_2050_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( nat2(Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_2051_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( nat2(I2) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_2052_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,nat2(W)),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_2053_nat__neg__numeral,axiom,
    ! [K: num] : nat2(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_2054_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,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,nat2(Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_2055_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_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_2056_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(minus_minus_nat(numeral_numeral_nat(V)),numeral_numeral_nat(V3)) = nat2(minus_minus_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_2057_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y2: int] :
      ( ( aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),N) = nat2(Y2) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y2 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_2058_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,N: nat] :
      ( ( nat2(Y2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),N) )
    <=> ( Y2 = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_2059_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,nat2(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_2060_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)),nat2(Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_2061_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(minus_minus_nat(numeral_numeral_nat(V)),one_one_nat) = nat2(minus_minus_int(aa_num_int(numeral_numeral_int,V),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_2062_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(power_power_nat(numeral_numeral_nat(X)),N)),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_2063_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,nat2(A)),aa_nat_nat(power_power_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_2064_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(power_power_nat(numeral_numeral_nat(X)),N)),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_2065_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,nat2(A)),aa_nat_nat(power_power_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_2066_nat__numeral__as__int,axiom,
    ! [X2: num] : numeral_numeral_nat(X2) = nat2(aa_num_int(numeral_numeral_int,X2)) ).

% nat_numeral_as_int
tff(fact_2067_nat__mono,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,nat2(X)),nat2(Y2))) ) ).

% nat_mono
tff(fact_2068_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: 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),Z5))
       => ( ( nat2(Z) = nat2(Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2069_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_nat_bool(P,nat2(X4))) ) ) ).

% all_nat
tff(fact_2070_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
    <=> ? [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
          & pp(aa_nat_bool(P,nat2(X4))) ) ) ).

% ex_nat
tff(fact_2071_nat__one__as__int,axiom,
    one_one_nat = nat2(one_one_int) ).

% nat_one_as_int
tff(fact_2072_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se4205575877204974255it_nat(M),N) = nat2(bit_se4203085406695923979it_int(M,aa_nat_int(semiri1314217659103216013at_int,N))) ).

% unset_bit_nat_def
tff(fact_2073_nat__mask__eq,axiom,
    ! [N: nat] : nat2(aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% nat_mask_eq
tff(fact_2074_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,nat2(W)),nat2(Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_2075_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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_2076_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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_2077_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,nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_2078_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = nat2(Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_2079_nat__int__add,axiom,
    ! [A: nat,B: nat] : nat2(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(plus_plus_nat(A),B) ).

% nat_int_add
tff(fact_2080_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : nat2(aa_int_int(abs_abs_int,times_times_int(W,Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,nat2(aa_int_int(abs_abs_int,W))),nat2(aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_2081_nat__plus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(plus_plus_nat(X2),Xa) = nat2(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X2),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_2082_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(M),N) = nat2(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% or_nat_def
tff(fact_2083_nat__times__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Xa) = nat2(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X2),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_2084_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,nat2(archim7802044766580827645g_real(X))))) ).

% real_nat_ceiling_ge
tff(fact_2085_nat__div__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(divide_divide_nat(X2),Xa) = nat2(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X2),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_2086_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_2087_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,nat2(W)),nat2(Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_2088_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,nat2(W)),nat2(Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_2089_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = 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_2090_nat__eq__iff,axiom,
    ! [W: int,M: 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_2091_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),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_2092_nat__add__distrib,axiom,
    ! [Z: int,Z5: 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),Z5))
       => ( nat2(plus_plus_int(Z,Z5)) = aa_nat_nat(plus_plus_nat(nat2(Z)),nat2(Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_2093_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( nat2(times_times_int(Z,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,nat2(Z)),nat2(Z5)) ) ) ).

% nat_mult_distrib
tff(fact_2094_Suc__as__int,axiom,
    ! [X2: nat] : aa_nat_nat(suc,X2) = nat2(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X2),one_one_int)) ).

% Suc_as_int
tff(fact_2095_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z))
       => ( nat2(minus_minus_int(Z,Z5)) = aa_nat_nat(minus_minus_nat(nat2(Z)),nat2(Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_2096_nat__diff__distrib_H,axiom,
    ! [X: int,Y2: 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),Y2))
       => ( nat2(minus_minus_int(X,Y2)) = aa_nat_nat(minus_minus_nat(nat2(X)),nat2(Y2)) ) ) ) ).

% nat_diff_distrib'
tff(fact_2097_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,nat2(aa_int_int(abs_abs_int,plus_plus_int(K,L)))),aa_nat_nat(plus_plus_nat(nat2(aa_int_int(abs_abs_int,K))),nat2(aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_2098_nat__div__distrib,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( nat2(divide_divide_int(X,Y2)) = aa_nat_nat(divide_divide_nat(nat2(X)),nat2(Y2)) ) ) ).

% nat_div_distrib
tff(fact_2099_nat__div__distrib_H,axiom,
    ! [Y2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y2))
     => ( nat2(divide_divide_int(X,Y2)) = aa_nat_nat(divide_divide_nat(nat2(X)),nat2(Y2)) ) ) ).

% nat_div_distrib'
tff(fact_2100_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( nat2(archim6058952711729229775r_real(X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_2101_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))
     => ( nat2(aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(power_power_nat(nat2(Z)),N) ) ) ).

% nat_power_eq
tff(fact_2102_nat__mod__distrib,axiom,
    ! [X: int,Y2: 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),Y2))
       => ( nat2(modulo_modulo_int(X,Y2)) = aa_nat_nat(modulo_modulo_nat(nat2(X)),nat2(Y2)) ) ) ) ).

% nat_mod_distrib
tff(fact_2103_div__abs__eq__div__nat,axiom,
    ! [K: int,L: 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(divide_divide_nat(nat2(aa_int_int(abs_abs_int,K))),nat2(aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_2104_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))))
       => ( nat2(archim6058952711729229775r_real(X)) = N ) ) ) ).

% floor_eq3
tff(fact_2105_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),nat2(archim6058952711729229775r_real(A)))) ) ).

% le_nat_floor
tff(fact_2106_nat__2,axiom,
    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_2107_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_2108_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,nat2(Z)) = nat2(plus_plus_int(one_one_int,Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_2109_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,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_2110_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( nat2(times_times_int(Z,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,nat2(aa_int_int(uminus_uminus_int,Z))),nat2(aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_2111_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( nat2(aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(minus_minus_nat(B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( nat2(aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(minus_minus_nat(A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_2112_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))))
       => ( nat2(archim6058952711729229775r_real(X)) = N ) ) ) ).

% floor_eq4
tff(fact_2113_num_Osize__gen_I1_J,axiom,
    size_num(one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_2114_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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_2115_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( sgn_sgn_complex(Z) = 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))
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_2116_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( sgn_sgn_complex(Z) = cis(arg(Z)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),arg(Z)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,arg(Z)),pi)) ) ) ).

% Arg_correct
tff(fact_2117_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,numeral_numeral_nat(aa_num_num(bit0,one))),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_2118_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),ring_1_of_int_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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),ring_1_of_int_real(N)) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),nat2(aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_2119_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(times_times_real(aa_real_real(sgn_sgn_real,X)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_2120_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(aa_num_num(bit0,X23)) = aa_nat_nat(plus_plus_nat(size_num(X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_2121_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_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) )
           => ( 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) )
               => ( bit_se725231765392027082nd_int(K,L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( bit_se725231765392027082nd_int(K,L) = plus_plus_int(times_times_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),modulo_modulo_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_2122_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( arg(Z) = zero_zero_real ) )
      & ( ( Z != zero_zero_complex )
       => ( arg(Z) = fChoice_real(aTP_Lamm_ca(Z)) ) ) ) ).

% Arg_def
tff(fact_2123_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_2124_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_concat_bit(N,K),zero_zero_int) = bit_se2923211474154528505it_int(N,K) ).

% concat_bit_of_zero_2
tff(fact_2125_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),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_2126_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2127_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2925701944663578781it_nat(N),aa_nat_nat(suc,zero_zero_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_2128_and__minus__numerals_I2_J,axiom,
    ! [N: num] : 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_2129_and__minus__numerals_I6_J,axiom,
    ! [N: num] : 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_2130_and__minus__numerals_I5_J,axiom,
    ! [N: num] : 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_2131_and__minus__numerals_I1_J,axiom,
    ! [N: num] : 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_2132_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))
     => ( nat2(bit_se2923211474154528505it_int(N,K)) = aa_nat_nat(bit_se2925701944663578781it_nat(N),nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_2133_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(bit_se2925701944663578781it_nat(N),nat2(K)) = nat2(bit_se2923211474154528505it_int(N,K)) ) ) ).

% take_bit_nat_eq
tff(fact_2134_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: 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(bit_se2925701944663578781it_nat(M),Q2)),aa_nat_nat(bit_se2925701944663578781it_nat(N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2135_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(bit_se2925701944663578781it_nat(N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_2136_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : bit_se2923211474154528505it_int(N,minus_minus_int(bit_se2923211474154528505it_int(N,K),bit_se2923211474154528505it_int(N,L))) = bit_se2923211474154528505it_int(N,minus_minus_int(K,L)) ).

% take_bit_diff
tff(fact_2137_take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_se2923211474154528505it_int(N,aa_int_int(uminus_uminus_int,bit_se2923211474154528505it_int(N,K))) = bit_se2923211474154528505it_int(N,aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_2138_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : bit_se2923211474154528505it_int(N,times_times_int(bit_se2923211474154528505it_int(N,K),bit_se2923211474154528505it_int(N,L))) = bit_se2923211474154528505it_int(N,times_times_int(K,L)) ).

% take_bit_mult
tff(fact_2139_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B: int] : bit_concat_bit(N,bit_se2923211474154528505it_int(N,B)) = bit_concat_bit(N,B) ).

% concat_bit_take_bit_eq
tff(fact_2140_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S: int] :
      ( ( aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(bit_concat_bit(N,R2),S) )
    <=> ( ( bit_se2923211474154528505it_int(N,K) = bit_se2923211474154528505it_int(N,R2) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_2141_AND__lower,axiom,
    ! [X: int,Y2: 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),bit_se725231765392027082nd_int(X,Y2))) ) ).

% AND_lower
tff(fact_2142_AND__upper1,axiom,
    ! [X: int,Y2: 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,bit_se725231765392027082nd_int(X,Y2)),X)) ) ).

% AND_upper1
tff(fact_2143_AND__upper2,axiom,
    ! [Y2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,bit_se725231765392027082nd_int(X,Y2)),Y2)) ) ).

% AND_upper2
tff(fact_2144_AND__upper1_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y2),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,bit_se725231765392027082nd_int(Y2,Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_2145_AND__upper2_H,axiom,
    ! [Y2: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y2),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,bit_se725231765392027082nd_int(X,Y2)),Z)) ) ) ).

% AND_upper2'
tff(fact_2146_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: 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,bit_se2923211474154528505it_int(M,K)),bit_se2923211474154528505it_int(N,K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2147_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),bit_se2923211474154528505it_int(N,K))) ).

% take_bit_nonnegative
tff(fact_2148_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,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_2149_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,bit_se2923211474154528505it_int(N,K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_2150_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_2151_plus__and__or,axiom,
    ! [X: int,Y2: int] : plus_plus_int(bit_se725231765392027082nd_int(X,Y2),bit_se1409905431419307370or_int(X,Y2)) = plus_plus_int(X,Y2) ).

% plus_and_or
tff(fact_2152_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,bit_se725231765392027082nd_int(K,L)),K)) ) ).

% and_less_eq
tff(fact_2153_AND__upper1_H_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y2),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,bit_se725231765392027082nd_int(Y2,Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_2154_AND__upper2_H_H,axiom,
    ! [Y2: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y2),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,bit_se725231765392027082nd_int(X,Y2)),Z)) ) ) ).

% AND_upper2''
tff(fact_2155_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(N,K) != zero_zero_int )
     => ( bit_se2923211474154528505it_int(N,minus_minus_int(K,one_one_int)) = minus_minus_int(bit_se2923211474154528505it_int(N,K),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_2156_even__and__iff__int,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))),bit_se725231765392027082nd_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))),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)) ) ) ).

% even_and_iff_int
tff(fact_2157_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(N,K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( bit_se2923211474154528505it_int(N,plus_plus_int(K,one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_2158_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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(bit_se2925701944663578781it_nat(N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2159_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(bit_se2925701944663578781it_nat(N),M)),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_2160_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(bit_se2925701944663578781it_nat(N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2161_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(bit_se2925701944663578781it_nat(N),M) = aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).

% take_bit_nat_def
tff(fact_2162_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2163_take__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se2923211474154528505it_int(N,K) = modulo_modulo_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_int_def
tff(fact_2164_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(bit_se2925701944663578781it_nat(N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_2165_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : 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)))) = times_times_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_2166_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),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_2167_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2168_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: 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_2169_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)))
       => ( bit_se2923211474154528505it_int(N,K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2170_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se2923211474154528505it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = times_times_int(bit_se2923211474154528505it_int(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_2171_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(N,K) != minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N),one_one_int) )
     => ( bit_se2923211474154528505it_int(N,plus_plus_int(K,one_one_int)) = plus_plus_int(one_one_int,bit_se2923211474154528505it_int(N,K)) ) ) ).

% take_bit_incr_eq
tff(fact_2172_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,bit_se2923211474154528505it_int(N,K)),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_2173_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,plus_plus_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),bit_se2923211474154528505it_int(N,K))) ) ).

% take_bit_int_greater_eq
tff(fact_2174_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : bit_ri631733984087533419it_int(N,K) = minus_minus_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N),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_2175_and__int__rec,axiom,
    ! [K: int,L: int] : bit_se725231765392027082nd_int(K,L) = 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)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_2176_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: 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)),plus_plus_int(K,one_one_int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2177_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)))
       => ( bit_se2923211474154528505it_int(N,aa_int_int(uminus_uminus_int,K)) = 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_2178_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se2923211474154528505it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = plus_plus_int(times_times_int(bit_se2923211474154528505it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_2179_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : 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)))) = plus_plus_int(times_times_int(bit_se2923211474154528505it_int(N,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_2180_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( cis(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_2181_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = numeral_numeral_nat(K) ).

% pred_numeral_inc
tff(fact_2182_and__nat__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit0,Y2))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_2183_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se727722235901077358nd_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_2184_and__nat__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit1,Y2))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_2185_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se727722235901077358nd_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_2186_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_2187_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_2188_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_2189_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_2190_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,inc(X3))) )
       => pp(aa_num_bool(P,X)) ) ) ).

% num_induct
tff(fact_2191_add__inc,axiom,
    ! [X: num,Y2: num] : plus_plus_num(X,inc(Y2)) = inc(plus_plus_num(X,Y2)) ).

% add_inc
tff(fact_2192_inc_Osimps_I1_J,axiom,
    inc(one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_2193_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_2194_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa_num_num(bit1,X)) = aa_num_num(bit0,inc(X)) ).

% inc.simps(3)
tff(fact_2195_add__One,axiom,
    ! [X: num] : plus_plus_num(X,one) = inc(X) ).

% add_One
tff(fact_2196_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_2197_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_2198_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(M),N) = nat2(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% and_nat_def
tff(fact_2199_mult__inc,axiom,
    ! [X: num,Y2: num] : times_times_num(X,inc(Y2)) = plus_plus_num(times_times_num(X,Y2),X) ).

% mult_inc
tff(fact_2200_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,aa_real_real(times_times_real(X),pi)) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_2201_sin__integer__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_2202_cos__integer__2pi,axiom,
    ! [N: real] :
      ( member_real(N,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_2203_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(bit_se727722235901077358nd_nat(M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(bit_se727722235901077358nd_nat(M),N) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_2204_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(M),N) = aa_nat_nat(plus_plus_nat(zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_2205_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))) )
       => ( 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))) )
       => ( bit_se725231765392027082nd_int(K,L) = 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)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_2206_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y2: int] :
      ( ( bit_se725231765392027082nd_int(X,Xa2) = Y2 )
     => ( ( ( 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(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
         => ( Y2 = 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))),Xa2))))) ) )
        & ( ~ ( 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(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
         => ( Y2 = 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))),Xa2)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(divide_divide_int(X,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(Xa2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_2207_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_ri631733984087533419it_int(N,K) = minus_minus_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N),K),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_2208_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : bit_se2923211474154528505it_int(aa_nat_nat(suc,N),K) = plus_plus_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))),bit_se2923211474154528505it_int(N,K)) ).

% take_bit_Suc_from_most
tff(fact_2209_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),bit_ri631733984087533419it_int(N,K)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_2210_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2211_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_2212_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_2213_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)))),numeral_numeral_nat(N)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),pred_numeral(N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_2214_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)))),numeral_numeral_nat(N)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),pred_numeral(N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_2215_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(bit_se725231765392027082nd_int(K,L)),N))
    <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).

% bit_and_int_iff
tff(fact_2216_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(bit_se1409905431419307370or_int(K,L)),N))
    <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
        | pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).

% bit_or_int_iff
tff(fact_2217_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_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_2218_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),bit_se2923211474154528505it_int(M,K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_2219_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(minus_minus_nat(N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_2220_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),plus_plus_int(one_one_int,N)))
     => ( set_or1266510415728281911st_int(M,plus_plus_int(one_one_int,N)) = insert_int(plus_plus_int(one_one_int,N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2221_simp__from__to,axiom,
    ! [J: int,I2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( set_or1266510415728281911st_int(I2,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( set_or1266510415728281911st_int(I2,J) = insert_int(I2,set_or1266510415728281911st_int(plus_plus_int(I2,one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_2222_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: 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_2223_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),M3))
             => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),M3))
              <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N3)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(minus_minus_nat(N3),one_one_nat)))
              <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N3)) ) ) ) ).

% int_bit_bound
tff(fact_2224_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% bit_int_def
tff(fact_2225_set__bit__eq,axiom,
    ! [N: nat,K: int] : bit_se7879613467334960850it_int(N,K) = plus_plus_int(K,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_2226_unset__bit__eq,axiom,
    ! [N: nat,K: int] : bit_se4203085406695923979it_int(N,K) = minus_minus_int(K,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_2227_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y2: int] :
      ( ( bit_se725231765392027082nd_int(X,Xa2) = Y2 )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2))
       => ~ ( ( ( ( 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(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => ( Y2 = 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))),Xa2))))) ) )
              & ( ~ ( 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(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => ( Y2 = 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))),Xa2)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(divide_divide_int(X,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(Xa2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_2228_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,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))) )
         => ( 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))) )
         => ( bit_se725231765392027082nd_int(K,L) = 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)))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_2229_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_2230_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_2231_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,K)) = insert_nat(K,set_ord_lessThan_nat(K)) ).

% lessThan_Suc
tff(fact_2232_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost_nat(aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),set_ord_atMost_nat(K)) ).

% atMost_Suc
tff(fact_2233_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)),numeral_numeral_nat(N))) ).

% not_bit_Suc_0_numeral
tff(fact_2234_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% atLeast0_atMost_Suc
tff(fact_2235_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,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = set_or1269000886237332187st_nat(M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_2236_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)))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2237_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))
     => ( set_or1269000886237332187st_nat(M,N) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2238_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan_nat(numeral_numeral_nat(K)) = insert_nat(pred_numeral(K),set_ord_lessThan_nat(pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_2239_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost_nat(numeral_numeral_nat(K)) = insert_nat(numeral_numeral_nat(K),set_ord_atMost_nat(pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_2240_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_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_2241_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N) = minus_minus_set_nat(set_ord_atMost_nat(N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_2242_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(M),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)))) ) ).

% bit_nat_def
tff(fact_2243_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ member_nat(N,nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(plus_plus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2244_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L2: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,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,divide_divide_int(K2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),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_2245_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I3,J2))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,plus_plus_int(I3,one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_2246_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa2: num,Y2: num] :
      ( ( bit_or_not_num_neg(X,Xa2) = Y2 )
     => ( accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y2 = one )
               => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [M5: num] :
                  ( ( Xa2 = aa_num_num(bit0,M5) )
                 => ( ( Y2 = aa_num_num(bit1,M5) )
                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit0,M5))) ) ) )
           => ( ( ( X = one )
               => ! [M5: num] :
                    ( ( Xa2 = aa_num_num(bit1,M5) )
                   => ( ( Y2 = aa_num_num(bit1,M5) )
                     => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit1,M5))) ) ) )
             => ( ! [N3: num] :
                    ( ( X = aa_num_num(bit0,N3) )
                   => ( ( Xa2 = one )
                     => ( ( Y2 = aa_num_num(bit0,one) )
                       => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N3),one)) ) ) )
               => ( ! [N3: num] :
                      ( ( X = aa_num_num(bit0,N3) )
                     => ! [M5: num] :
                          ( ( Xa2 = aa_num_num(bit0,M5) )
                         => ( ( Y2 = bitM(bit_or_not_num_neg(N3,M5)) )
                           => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N3),aa_num_num(bit0,M5))) ) ) )
                 => ( ! [N3: num] :
                        ( ( X = aa_num_num(bit0,N3) )
                       => ! [M5: num] :
                            ( ( Xa2 = aa_num_num(bit1,M5) )
                           => ( ( Y2 = aa_num_num(bit0,bit_or_not_num_neg(N3,M5)) )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N3),aa_num_num(bit1,M5))) ) ) )
                   => ( ! [N3: num] :
                          ( ( X = aa_num_num(bit1,N3) )
                         => ( ( Xa2 = one )
                           => ( ( Y2 = one )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N3),one)) ) ) )
                     => ( ! [N3: num] :
                            ( ( X = aa_num_num(bit1,N3) )
                           => ! [M5: num] :
                                ( ( Xa2 = aa_num_num(bit0,M5) )
                               => ( ( Y2 = bitM(bit_or_not_num_neg(N3,M5)) )
                                 => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N3),aa_num_num(bit0,M5))) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X = aa_num_num(bit1,N3) )
                             => ! [M5: num] :
                                  ( ( Xa2 = aa_num_num(bit1,M5) )
                                 => ( ( Y2 = bitM(bit_or_not_num_neg(N3,M5)) )
                                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N3),aa_num_num(bit1,M5))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_2247_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_cb(D))) ).

% int_ge_less_than2_def
tff(fact_2248_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_cc(D))) ).

% int_ge_less_than_def
tff(fact_2249_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y2: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y2) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y2)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv) )
               => ( ~ pp(Y2)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv))) ) )
           => ( ! [Uu: bool] :
                  ( ( X = vEBT_Leaf(Uu,fTrue) )
                 => ( ~ pp(Y2)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu,fTrue))) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy2) )
                   => ( pp(Y2)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,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(Y2)
                       => ~ 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_2250_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))
       => ( ! [Uv: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv))) )
         => ( ! [Uu: bool] :
                ( ( X = vEBT_Leaf(Uu,fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu,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_2251_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))) )
         => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_2252_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_2253_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_2254_upto__aux__rec,axiom,
    ! [J: int,I2: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( upto_aux(I2,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( upto_aux(I2,J,Js) = upto_aux(I2,minus_minus_int(J,one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_2255_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_2256_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se6528837805403552850or_nat(numeral_numeral_nat(aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_2257_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se6528837805403552850or_nat(numeral_numeral_nat(aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_2258_xor__nat__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit1,Y2))) = numeral_numeral_nat(aa_num_num(bit0,Y2)) ).

% xor_nat_numerals(2)
tff(fact_2259_xor__nat__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(aa_num_num(bit0,Y2))) = numeral_numeral_nat(aa_num_num(bit1,Y2)) ).

% xor_nat_numerals(1)
tff(fact_2260_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(bit_se6528837805403552850or_nat(M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(bit_se6528837805403552850or_nat(M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(bit_se6528837805403552850or_nat(M),N) = aa_nat_nat(plus_plus_nat(aa_nat_nat(modulo_modulo_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(modulo_modulo_nat(N),numeral_numeral_nat(aa_num_num(bit0,one))))),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_2261_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(M),N) = aa_nat_nat(plus_plus_nat(zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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,numeral_numeral_nat(aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_2262_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I2,J))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
         => ( upto(I2,J) = cons_int(I2,upto(plus_plus_int(I2,one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
         => ( upto(I2,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_2263_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y2: list_int] :
      ( ( upto(X,Xa2) = Y2 )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa2))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y2 = cons_int(X,upto(plus_plus_int(X,one_one_int),Xa2)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y2 = nil_int ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% upto.pelims
tff(fact_2264_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_2265_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),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_2266_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2267_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa_int_int(bit_concat_bit(N,zero_zero_int),L) = bit_se545348938243370406it_int(N,L) ).

% concat_bit_of_zero_1
tff(fact_2268_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),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_2269_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2270_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2)) ) ).

% upto_Nil
tff(fact_2271_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int = upto(I2,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2)) ) ).

% upto_Nil2
tff(fact_2272_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
     => ( upto(I2,J) = nil_int ) ) ).

% upto_empty
tff(fact_2273_upto__single,axiom,
    ! [I2: int] : upto(I2,I2) = cons_int(I2,nil_int) ).

% upto_single
tff(fact_2274_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,plus_plus_int(I2,aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I2,J)),K) = plus_plus_int(I2,aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_2275_length__upto,axiom,
    ! [I2: int,J: int] : size_size_list_int(upto(I2,J)) = nat2(plus_plus_int(minus_minus_int(J,I2),one_one_int)) ).

% length_upto
tff(fact_2276_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(bit_se547839408752420682it_nat(N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_2277_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(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_2278_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(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_2279_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(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_2280_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(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_2281_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(bit_se6526347334894502574or_int(K,L)),N))
    <=> ~ ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
        <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_2282_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se2159334234014336723it_int(N,K) = bit_se6526347334894502574or_int(K,bit_se545348938243370406it_int(N,one_one_int)) ).

% flip_bit_int_def
tff(fact_2283_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or1266510415728281911st_int(I2,J) = set_int2(upto(I2,J)) ).

% atLeastAtMost_upto
tff(fact_2284_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(bit_se547839408752420682it_nat(N),nat2(K)) = nat2(bit_se545348938243370406it_int(N,K)) ).

% push_bit_nat_eq
tff(fact_2285_upto__code,axiom,
    ! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil_int) ).

% upto_code
tff(fact_2286_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list_int] : upto_aux(I2,J,Js) = append_int(upto(I2,J),Js) ).

% upto_aux_def
tff(fact_2287_XOR__lower,axiom,
    ! [X: int,Y2: 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),Y2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),bit_se6526347334894502574or_int(X,Y2))) ) ) ).

% XOR_lower
tff(fact_2288_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se7882103937844011126it_nat(M),N) = aa_nat_nat(bit_se1412395901928357646or_nat(N),aa_nat_nat(bit_se547839408752420682it_nat(M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_2289_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se2161824704523386999it_nat(M),N) = aa_nat_nat(bit_se6528837805403552850or_nat(N),aa_nat_nat(bit_se547839408752420682it_nat(M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_2290_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_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(minus_minus_nat(N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_2291_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(M),N) = nat2(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% xor_nat_def
tff(fact_2292_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(bit_se547839408752420682it_nat(M),Q2)),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,Q2),aa_nat_nat(minus_minus_nat(N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_2293_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = plus_plus_int(bit_se2923211474154528505it_int(N,K),bit_se545348938243370406it_int(N,L)) ).

% concat_bit_eq
tff(fact_2294_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I2,K) = append_int(upto(I2,J),upto(plus_plus_int(J,one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_2295_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I2,K) = append_int(upto(I2,minus_minus_int(J,one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_2296_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = bit_se1409905431419307370or_int(bit_se2923211474154528505it_int(N,K),bit_se545348938243370406it_int(N,L)) ).

% concat_bit_def
tff(fact_2297_set__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se7879613467334960850it_int(N,K) = bit_se1409905431419307370or_int(K,bit_se545348938243370406it_int(N,one_one_int)) ).

% set_bit_int_def
tff(fact_2298_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( upto(I2,J) = cons_int(I2,upto(plus_plus_int(I2,one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_2299_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
       => ( upto(I2,J) = cons_int(I2,upto(plus_plus_int(I2,one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
       => ( upto(I2,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_2300_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y2: list_int] :
      ( ( upto(X,Xa2) = Y2 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y2 = cons_int(X,upto(plus_plus_int(X,one_one_int),Xa2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y2 = nil_int ) ) ) ) ).

% upto.elims
tff(fact_2301_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( upto(I2,J) = append_int(upto(I2,minus_minus_int(J,one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_2302_push__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se545348938243370406it_int(N,K) = times_times_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_int_def
tff(fact_2303_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(bit_se547839408752420682it_nat(N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_2304_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I2,K) = append_int(upto(I2,minus_minus_int(J,one_one_int)),cons_int(J,upto(plus_plus_int(J,one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_2305_push__bit__minus__one,axiom,
    ! [N: nat] : 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_2306_XOR__upper,axiom,
    ! [X: int,N: nat,Y2: 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,Y2),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,bit_se6526347334894502574or_int(X,Y2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_2307_xor__int__rec,axiom,
    ! [K: int,L: int] : bit_se6526347334894502574or_int(K,L) = plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_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))))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se6526347334894502574or_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_2308_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( bit_se6526347334894502574or_int(K,L) = bit_ri7919022796975470100ot_int(L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( bit_se6526347334894502574or_int(K,L) = bit_ri7919022796975470100ot_int(K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( bit_se6526347334894502574or_int(K,L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( bit_se6526347334894502574or_int(K,L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( bit_se6526347334894502574or_int(K,L) = plus_plus_int(aa_int_int(abs_abs_int,minus_minus_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),modulo_modulo_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se6526347334894502574or_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),divide_divide_int(L,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_2309_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( finite_finite_nat(A2)
     => ( ~ member_nat(N,A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A2)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_2310_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_2311_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_2312_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_2313_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => finite_finite_nat(vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_2314_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2315_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_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_2316_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : 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))) = bit_ri7919022796975470100ot_int(bit_se1409905431419307370or_int(minus_minus_int(aa_num_int(numeral_numeral_int,M),one_one_int),minus_minus_int(aa_num_int(numeral_numeral_int,N),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_2317_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : 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))) = bit_ri7919022796975470100ot_int(bit_se725231765392027082nd_int(minus_minus_int(aa_num_int(numeral_numeral_int,M),one_one_int),minus_minus_int(aa_num_int(numeral_numeral_int,N),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_2318_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(bit_ri7919022796975470100ot_int(K)),N))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% bit_not_int_iff
tff(fact_2319_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I2: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cd(P),I2))) ).

% finite_M_bounded_by_nat
tff(fact_2320_finite__nat__set__iff__bounded,axiom,
    ! [N2: set_nat] :
      ( finite_finite_nat(N2)
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( member_nat(X4,N2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),M6)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_2321_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( member_nat(X3,N2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),N)) )
     => finite_finite_nat(N2) ) ).

% bounded_nat_set_is_finite
tff(fact_2322_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set_nat] :
      ( finite_finite_nat(N2)
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( member_nat(X4,N2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M6)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_2323_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),aa_nat_nat(F,N3)))
     => finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ce(F),U))) ) ).

% finite_less_ub
tff(fact_2324_or__int__def,axiom,
    ! [K: int,L: int] : bit_se1409905431419307370or_int(K,L) = bit_ri7919022796975470100ot_int(bit_se725231765392027082nd_int(bit_ri7919022796975470100ot_int(K),bit_ri7919022796975470100ot_int(L))) ).

% or_int_def
tff(fact_2325_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: bool,Uv2: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu2,Uv2)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_2326_not__int__def,axiom,
    ! [K: int] : bit_ri7919022796975470100ot_int(K) = minus_minus_int(aa_int_int(uminus_uminus_int,K),one_one_int) ).

% not_int_def
tff(fact_2327_and__not__numerals_I1_J,axiom,
    bit_se725231765392027082nd_int(one_one_int,bit_ri7919022796975470100ot_int(one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_2328_or__not__numerals_I1_J,axiom,
    bit_se1409905431419307370or_int(one_one_int,bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(zero_zero_int) ).

% or_not_numerals(1)
tff(fact_2329_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se4203085406695923979it_int(N,K) = bit_se725231765392027082nd_int(K,bit_ri7919022796975470100ot_int(bit_se545348938243370406it_int(N,one_one_int))) ).

% unset_bit_int_def
tff(fact_2330_xor__int__def,axiom,
    ! [K: int,L: int] : bit_se6526347334894502574or_int(K,L) = bit_se1409905431419307370or_int(bit_se725231765392027082nd_int(K,bit_ri7919022796975470100ot_int(L)),bit_se725231765392027082nd_int(bit_ri7919022796975470100ot_int(K),L)) ).

% xor_int_def
tff(fact_2331_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cf,M))) ) ).

% finite_divisors_nat
tff(fact_2332_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( ord_less_eq_set_nat(N2,set_or1269000886237332187st_nat(zero_zero_nat,N))
     => finite_finite_nat(N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2333_not__int__div__2,axiom,
    ! [K: int] : divide_divide_int(bit_ri7919022796975470100ot_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = bit_ri7919022796975470100ot_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_2334_even__not__iff__int,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))),bit_ri7919022796975470100ot_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_not_iff_int
tff(fact_2335_and__not__numerals_I2_J,axiom,
    ! [N: num] : bit_se725231765392027082nd_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_2336_and__not__numerals_I4_J,axiom,
    ! [M: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)),bit_ri7919022796975470100ot_int(one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_2337_or__not__numerals_I4_J,axiom,
    ! [M: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(one_one_int) ).

% or_not_numerals(4)
tff(fact_2338_or__not__numerals_I2_J,axiom,
    ! [N: num] : bit_se1409905431419307370or_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(2)
tff(fact_2339_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(bit_ri7919022796975470100ot_int(minus_minus_int(K,one_one_int))),N)) ) ).

% bit_minus_int_iff
tff(fact_2340_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,N))) ).

% int_numeral_or_not_num_neg
tff(fact_2341_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : bit_se1409905431419307370or_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,bit_or_not_num_neg(N,M))) ).

% int_numeral_not_or_num_neg
tff(fact_2342_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,N)) = aa_int_int(uminus_uminus_int,bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N)))) ).

% numeral_or_not_num_eq
tff(fact_2343_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(5)
tff(fact_2344_and__not__numerals_I7_J,axiom,
    ! [M: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M)),bit_ri7919022796975470100ot_int(one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_2345_or__not__numerals_I3_J,axiom,
    ! [N: num] : bit_se1409905431419307370or_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(3)
tff(fact_2346_and__not__numerals_I3_J,axiom,
    ! [N: num] : bit_se725231765392027082nd_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_2347_or__not__numerals_I7_J,axiom,
    ! [M: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M)),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(zero_zero_int) ).

% or_not_numerals(7)
tff(fact_2348_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(6)
tff(fact_2349_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(9)
tff(fact_2350_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N)))) ).

% or_not_numerals(6)
tff(fact_2351_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( finite_finite_nat(A2)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2)))
      <=> ~ member_nat(zero_zero_nat,A2) ) ) ).

% even_set_encode_iff
tff(fact_2352_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_2353_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))))) ).

% and_not_numerals(8)
tff(fact_2354_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_2355_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_2356_not__int__rec,axiom,
    ! [K: int] : bit_ri7919022796975470100ot_int(K) = 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)),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),bit_ri7919022796975470100ot_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_2357_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,K))) ).

% finite_Collect_le_nat
tff(fact_2358_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ak,K))) ).

% finite_Collect_less_nat
tff(fact_2359_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ch(A),B))) ).

% finite_interval_int1
tff(fact_2360_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ci(A),B))) ).

% finite_interval_int2
tff(fact_2361_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_cj(A),B))) ).

% finite_interval_int3
tff(fact_2362_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_bb(N),C))) ) ).

% finite_nth_roots
tff(fact_2363_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( finite_finite_nat(S3)
    <=> ? [K3: nat] : ord_less_eq_set_nat(S3,set_ord_atMost_nat(K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_2364_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(real_V4546457046886955230omplex(aa_real_real(aa_nat_fun_real_real(root,N),real_V1022390504157884413omplex(C)))),cis(aa_real_real(divide_divide_real(arg(C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aTP_Lamm_bc(N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_ck(C),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_2365_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( finite_finite_nat(S3)
     => ? [K2: nat] : ord_less_eq_set_nat(S3,set_ord_lessThan_nat(K2)) ) ).

% finite_nat_bounded
tff(fact_2366_real__root__zero,axiom,
    ! [N: nat] : aa_real_real(aa_nat_fun_real_real(root,N),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_2367_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_2368_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_2369_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y2: 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),Y2) )
      <=> ( X = Y2 ) ) ) ).

% real_root_eq_iff
tff(fact_2370_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_2371_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y2: 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),Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2)) ) ) ).

% real_root_less_iff
tff(fact_2372_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y2: 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),Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y2)) ) ) ).

% real_root_le_iff
tff(fact_2373_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_2374_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_2375_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_2376_real__root__gt__0__iff,axiom,
    ! [N: nat,Y2: 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),Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y2)) ) ) ).

% real_root_gt_0_iff
tff(fact_2377_real__root__ge__0__iff,axiom,
    ! [N: nat,Y2: 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),Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y2)) ) ) ).

% real_root_ge_0_iff
tff(fact_2378_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_2379_real__root__gt__1__iff,axiom,
    ! [N: nat,Y2: 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),Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y2)) ) ) ).

% real_root_gt_1_iff
tff(fact_2380_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_2381_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_2382_real__root__ge__1__iff,axiom,
    ! [N: nat,Y2: 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),Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y2)) ) ) ).

% real_root_ge_1_iff
tff(fact_2383_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(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_2384_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_2385_real__root__divide,axiom,
    ! [N: nat,X: real,Y2: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y2)) ).

% real_root_divide
tff(fact_2386_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_nat_fun_real_real(root,M),X)) ).

% real_root_commute
tff(fact_2387_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),X) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_mult_exp
tff(fact_2388_real__root__mult,axiom,
    ! [N: nat,X: real,Y2: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y2)) ).

% real_root_mult
tff(fact_2389_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_2390_real__root__pos__pos__le,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,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ).

% real_root_pos_pos_le
tff(fact_2391_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y2: 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),Y2))
       => 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),Y2))) ) ) ).

% real_root_less_mono
tff(fact_2392_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y2: 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),Y2))
       => 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),Y2))) ) ) ).

% real_root_le_mono
tff(fact_2393_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(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).

% real_root_power
tff(fact_2394_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_2395_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_2396_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_2397_real__root__strict__decreasing,axiom,
    ! [N: nat,N2: 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),N2))
       => ( 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,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_2398_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,numeral_numeral_nat(aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_2399_root__abs__power,axiom,
    ! [N: nat,Y2: 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(aa_real_fun_nat_real(power_power_real,Y2),N))) = aa_real_real(abs_abs_real,Y2) ) ) ).

% root_abs_power
tff(fact_2400_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_2401_real__root__strict__increasing,axiom,
    ! [N: nat,N2: 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),N2))
       => ( 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,N2),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_2402_real__root__decreasing,axiom,
    ! [N: nat,N2: 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),N2))
       => ( 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,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_2403_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(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_2404_real__root__pos__unique,axiom,
    ! [N: nat,Y2: 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),Y2))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y2 ) ) ) ) ).

% real_root_pos_unique
tff(fact_2405_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(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_2406_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ).

% odd_real_root_pow
tff(fact_2407_odd__real__root__unique,axiom,
    ! [N: nat,Y2: real,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),N) = X )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y2 ) ) ) ).

% odd_real_root_unique
tff(fact_2408_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_2409_real__root__increasing,axiom,
    ! [N: nat,N2: 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),N2))
       => ( 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,N2),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_2410_root__sgn__power,axiom,
    ! [N: nat,Y2: 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(times_times_real(aa_real_real(sgn_sgn_real,Y2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y2)),N))) = Y2 ) ) ).

% root_sgn_power
tff(fact_2411_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(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_2412_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_2413_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(log(B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_2414_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(log(aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B),X)) ) ) ) ).

% log_base_root
tff(fact_2415_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))
         => ! [Y4: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),N)) = X )
             => pp(aa_real_bool(P,Y4)) ) ) ) ) ).

% split_root
tff(fact_2416_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ! [M6: int] :
        ? [N4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M6),aa_int_int(abs_abs_int,N4)))
          & member_int(N4,S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2417_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ finite_finite_nat(S3)
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N4))
          & member_nat(N4,S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_2418_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M5))
         => ? [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N6))
              & member_nat(N6,S3) ) )
     => ~ finite_finite_nat(S3) ) ).

% unbounded_k_infinite
tff(fact_2419_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ finite_finite_nat(S3)
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
          & member_nat(N4,S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_2420_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_2421_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( finite_finite_nat(S3)
    <=> ? [K3: nat] : ord_less_eq_set_nat(S3,set_ord_lessThan_nat(K3)) ) ).

% finite_nat_iff_bounded
tff(fact_2422_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_al),set_or4665077453230672383an_nat(M,N)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(minus_minus_nat(N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(minus_minus_nat(M),one_one_nat)))),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_2423_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M6))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),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(X6,M6)),aa_nat_real(X6,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_2424_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_2425_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_2426_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N))
          & pp(aa_nat_bool(P,M6)) )
    <=> ? [X4: nat] :
          ( member_nat(X4,set_or4665077453230672383an_nat(zero_zero_nat,N))
          & pp(aa_nat_bool(P,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_2427_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N))
         => pp(aa_nat_bool(P,M6)) )
    <=> ! [X4: nat] :
          ( member_nat(X4,set_or4665077453230672383an_nat(zero_zero_nat,N))
         => pp(aa_nat_bool(P,X4)) ) ) ).

% all_nat_less_eq
tff(fact_2428_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_2429_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% atLeast0_lessThan_Suc
tff(fact_2430_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( ord_less_eq_set_nat(N2,set_or4665077453230672383an_nat(zero_zero_nat,N))
     => finite_finite_nat(N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_2431_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(M,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_2432_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_2433_prod__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_2434_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),pred_numeral(K)))
       => ( set_or4665077453230672383an_nat(M,numeral_numeral_nat(K)) = insert_nat(pred_numeral(K),set_or4665077453230672383an_nat(M,pred_numeral(K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),pred_numeral(K)))
       => ( set_or4665077453230672383an_nat(M,numeral_numeral_nat(K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_2435_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N) = minus_minus_set_nat(set_ord_lessThan_nat(N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_2436_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),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,I3)),aa_nat_nat(A,J2))) ) )
     => ( ! [I3: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),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,I3))) ) )
       => 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_cl(A),B)),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),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_2437_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,plus_plus_int(U,one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_2438_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or4662586982721622107an_int(I2,J) = set_int2(upto(I2,minus_minus_int(J,one_one_int))) ).

% atLeastLessThan_upto
tff(fact_2439_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_2440_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_cm(L),Qr) ).

% divmod_step_integer_def
tff(fact_2441_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_2442_card__Collect__less__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ak,N))) = N ).

% card_Collect_less_nat
tff(fact_2443_card__atMost,axiom,
    ! [U: nat] : finite_card_nat(set_ord_atMost_nat(U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_2444_card__Collect__le__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_2445_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(set_or1269000886237332187st_nat(L,U)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_2446_complex__Re__numeral,axiom,
    ! [V: num] : re(numera6690914467698888265omplex(V)) = numeral_numeral_real(V) ).

% complex_Re_numeral
tff(fact_2447_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = nat2(plus_plus_int(minus_minus_int(U,L),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_2448_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(divide1717551699836669952omplex(Z,numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(re(Z)),numeral_numeral_real(W)) ).

% Re_divide_numeral
tff(fact_2449_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% divmod_integer'_def
tff(fact_2450_less__eq__integer__code_I1_J,axiom,
    pp(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(1)
tff(fact_2451_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_2452_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_2453_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_2454_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_2455_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_2456_complex__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,re(X)),real_V1022390504157884413omplex(X))) ).

% complex_Re_le_cmod
tff(fact_2457_one__complex_Osimps_I1_J,axiom,
    re(one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_2458_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y2: complex] : re(plus_plus_complex(X,Y2)) = aa_real_real(plus_plus_real(re(X)),re(Y2)) ).

% plus_complex.simps(1)
tff(fact_2459_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X: complex] : re(real_V2046097035970521341omplex(R2,X)) = aa_real_real(times_times_real(R2),re(X)) ).

% scaleR_complex.simps(1)
tff(fact_2460_abs__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,re(X))),real_V1022390504157884413omplex(X))) ).

% abs_Re_le_cmod
tff(fact_2461_Re__csqrt,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),re(csqrt(Z)))) ).

% Re_csqrt
tff(fact_2462_card__less__Suc2,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ~ member_nat(zero_zero_nat,M7)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cn(M7),I2))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_co(M7),I2))) ) ) ).

% card_less_Suc2
tff(fact_2463_card__less__Suc,axiom,
    ! [M7: set_nat,I2: nat] :
      ( member_nat(zero_zero_nat,M7)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cn(M7),I2)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_co(M7),I2))) ) ) ).

% card_less_Suc
tff(fact_2464_card__less,axiom,
    ! [M7: set_nat,I2: nat] :
      ( member_nat(zero_zero_nat,M7)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_co(M7),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_2465_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_2466_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ord_less_eq_set_nat(A2,set_or4665077453230672383an_nat(K,aa_nat_nat(plus_plus_nat(K),finite_card_nat(A2))))
     => ( A2 = set_or4665077453230672383an_nat(K,aa_nat_nat(plus_plus_nat(K),finite_card_nat(A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_2467_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_2468_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,N: nat] :
      ( ord_less_eq_set_nat(N2,set_or4665077453230672383an_nat(zero_zero_nat,N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(N2)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_2469_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_al),set_or4665077453230672383an_nat(zero_zero_nat,finite_card_nat(S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_al),S3))) ).

% card_sum_le_nat_sum
tff(fact_2470_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_ck(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_2471_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_bc(N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_2472_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(real_V1022390504157884413omplex(Z)),re(Z))),zero_zero_real))
    <=> ( re(Z) = aa_real_real(uminus_uminus_real,real_V1022390504157884413omplex(Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_2473_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),A)) = re(power_power_complex(cis(A),N)) ).

% cos_n_Re_cis_pow_n
tff(fact_2474_csqrt_Ocode,axiom,
    ! [Z: complex] : csqrt(Z) = complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(times_times_real(aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,im(Z)),zero_zero_real),one_one_real),aa_real_real(sgn_sgn_real,im(Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_2475_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] : im(csqrt(Z)) = aa_real_real(times_times_real(aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,im(Z)),zero_zero_real),one_one_real),aa_real_real(sgn_sgn_real,im(Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_2476_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( code_integer_of_int(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,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 )
           => ( code_integer_of_int(K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( code_integer_of_int(K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_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))),code_integer_of_int(divide_divide_int(K,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),code_integer_of_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_2477_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( im(X) = zero_zero_real )
     => ( im(power_power_complex(X,N)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_2478_complex__Im__numeral,axiom,
    ! [V: num] : im(numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_2479_Im__i__times,axiom,
    ! [Z: complex] : im(aa_complex_complex(times_times_complex(imaginary_unit),Z)) = re(Z) ).

% Im_i_times
tff(fact_2480_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( im(X) = zero_zero_real )
     => ( re(power_power_complex(X,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),N) ) ) ).

% Re_power_real
tff(fact_2481_Re__i__times,axiom,
    ! [Z: complex] : re(aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,im(Z)) ).

% Re_i_times
tff(fact_2482_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(divide1717551699836669952omplex(Z,numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(im(Z)),numeral_numeral_real(W)) ).

% Im_divide_numeral
tff(fact_2483_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),re(X)))
       => ( csqrt(X) = real_V4546457046886955230omplex(aa_real_real(sqrt,re(X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_2484_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,im(X)),zero_zero_real))
        | ( ( im(X) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),re(X))) ) )
     => ( csqrt(uminus1482373934393186551omplex(X)) = aa_complex_complex(times_times_complex(imaginary_unit),csqrt(X)) ) ) ).

% csqrt_minus
tff(fact_2485_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,re(X)),zero_zero_real))
       => ( csqrt(X) = aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,aa_real_real(abs_abs_real,re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_2486_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_2487_one__complex_Osimps_I2_J,axiom,
    im(one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_2488_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : plus_p5714425477246183910nteger(code_integer_of_int(Xa2),code_integer_of_int(X)) = code_integer_of_int(plus_plus_int(Xa2,X)) ).

% plus_integer.abs_eq
tff(fact_2489_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_integer_of_int(Xa2)),code_integer_of_int(X)) = code_integer_of_int(times_times_int(Xa2,X)) ).

% times_integer.abs_eq
tff(fact_2490_one__integer__def,axiom,
    one_one_Code_integer = code_integer_of_int(one_one_int) ).

% one_integer_def
tff(fact_2491_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(ord_le3102999989581377725nteger(code_integer_of_int(Xa2),code_integer_of_int(X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_2492_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y2: complex] : im(plus_plus_complex(X,Y2)) = aa_real_real(plus_plus_real(im(X)),im(Y2)) ).

% plus_complex.simps(2)
tff(fact_2493_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X: complex] : im(real_V2046097035970521341omplex(R2,X)) = aa_real_real(times_times_real(R2),im(X)) ).

% scaleR_complex.simps(2)
tff(fact_2494_abs__Im__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,im(X))),real_V1022390504157884413omplex(X))) ).

% abs_Im_le_cmod
tff(fact_2495_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y2: complex] : im(aa_complex_complex(times_times_complex(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(re(X)),im(Y2))),aa_real_real(times_times_real(im(X)),re(Y2))) ).

% times_complex.simps(2)
tff(fact_2496_cmod__Im__le__iff,axiom,
    ! [X: complex,Y2: complex] :
      ( ( re(X) = re(Y2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real_V1022390504157884413omplex(X)),real_V1022390504157884413omplex(Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,im(X))),aa_real_real(abs_abs_real,im(Y2)))) ) ) ).

% cmod_Im_le_iff
tff(fact_2497_cmod__Re__le__iff,axiom,
    ! [X: complex,Y2: complex] :
      ( ( im(X) = im(Y2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real_V1022390504157884413omplex(X)),real_V1022390504157884413omplex(Y2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,re(X))),aa_real_real(abs_abs_real,re(Y2)))) ) ) ).

% cmod_Re_le_iff
tff(fact_2498_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y2: complex] : re(aa_complex_complex(times_times_complex(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(re(X)),re(Y2))),aa_real_real(times_times_real(im(X)),im(Y2))) ).

% times_complex.simps(1)
tff(fact_2499_plus__complex_Ocode,axiom,
    ! [X: complex,Y2: complex] : plus_plus_complex(X,Y2) = complex2(aa_real_real(plus_plus_real(re(X)),re(Y2)),aa_real_real(plus_plus_real(im(X)),im(Y2))) ).

% plus_complex.code
tff(fact_2500_scaleR__complex_Ocode,axiom,
    ! [R2: real,X: complex] : real_V2046097035970521341omplex(R2,X) = complex2(aa_real_real(times_times_real(R2),re(X)),aa_real_real(times_times_real(R2),im(X))) ).

% scaleR_complex.code
tff(fact_2501_csqrt__principal,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(csqrt(Z))))
      | ( ( re(csqrt(Z)) = zero_zero_real )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),im(csqrt(Z)))) ) ) ).

% csqrt_principal
tff(fact_2502_cmod__le,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real_V1022390504157884413omplex(Z)),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,re(Z))),aa_real_real(abs_abs_real,im(Z))))) ).

% cmod_le
tff(fact_2503_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),A)) = im(power_power_complex(cis(A),N)) ).

% sin_n_Im_cis_pow_n
tff(fact_2504_Re__exp,axiom,
    ! [Z: complex] : re(exp_complex(Z)) = aa_real_real(times_times_real(aa_real_real(exp_real,re(Z))),aa_real_real(cos_real,im(Z))) ).

% Re_exp
tff(fact_2505_Im__exp,axiom,
    ! [Z: complex] : im(exp_complex(Z)) = aa_real_real(times_times_real(aa_real_real(exp_real,re(Z))),aa_real_real(sin_real,im(Z))) ).

% Im_exp
tff(fact_2506_complex__eq,axiom,
    ! [A: complex] : A = plus_plus_complex(real_V4546457046886955230omplex(re(A)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(im(A)))) ).

% complex_eq
tff(fact_2507_times__complex_Ocode,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(times_times_complex(X),Y2) = complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(re(X)),re(Y2))),aa_real_real(times_times_real(im(X)),im(Y2))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(re(X)),im(Y2))),aa_real_real(times_times_real(im(X)),re(Y2)))) ).

% times_complex.code
tff(fact_2508_exp__eq__polar,axiom,
    ! [Z: complex] : exp_complex(Z) = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(exp_real,re(Z)))),cis(im(Z))) ).

% exp_eq_polar
tff(fact_2509_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_2510_Im__power2,axiom,
    ! [X: complex] : im(power_power_complex(X,numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),re(X))),im(X)) ).

% Im_power2
tff(fact_2511_Re__power2,axiom,
    ! [X: complex] : re(power_power_complex(X,numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_2512_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_2513_norm__complex__def,axiom,
    ! [Z: complex] : real_V1022390504157884413omplex(Z) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_2514_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(invers8013647133539491842omplex(X)) = aa_real_real(divide_divide_real(re(X)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_2515_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))))) ) ).

% complex_neq_0
tff(fact_2516_Re__divide,axiom,
    ! [X: complex,Y2: complex] : re(divide1717551699836669952omplex(X,Y2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(re(X)),re(Y2))),aa_real_real(times_times_real(im(X)),im(Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y2)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y2)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_2517_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( power_power_complex(W,numeral_numeral_nat(aa_num_num(bit0,one))) = Z )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(W)))
          | ( ( re(W) = zero_zero_real )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),im(W))) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_2518_csqrt__square,axiom,
    ! [B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(B)))
        | ( ( re(B) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),im(B))) ) )
     => ( csqrt(power_power_complex(B,numeral_numeral_nat(aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_2519_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(invers8013647133539491842omplex(X)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,im(X))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_2520_Im__divide,axiom,
    ! [X: complex,Y2: complex] : im(divide1717551699836669952omplex(X,Y2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(im(X)),re(Y2))),aa_real_real(times_times_real(re(X)),im(Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y2)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y2)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_2521_complex__abs__le__norm,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,re(Z))),aa_real_real(abs_abs_real,im(Z)))),aa_real_real(times_times_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),real_V1022390504157884413omplex(Z)))) ).

% complex_abs_le_norm
tff(fact_2522_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(re(Z)),real_V1022390504157884413omplex(Z))),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(im(Z)),real_V1022390504157884413omplex(Z))),numeral_numeral_nat(aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_2523_inverse__complex_Ocode,axiom,
    ! [X: complex] : invers8013647133539491842omplex(X) = complex2(aa_real_real(divide_divide_real(re(X)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,im(X))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_2524_Complex__divide,axiom,
    ! [X: complex,Y2: complex] : divide1717551699836669952omplex(X,Y2) = complex2(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(re(X)),re(Y2))),aa_real_real(times_times_real(im(X)),im(Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y2)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y2)),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(im(X)),re(Y2))),aa_real_real(times_times_real(re(X)),im(Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y2)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y2)),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_2525_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( im(divide1717551699836669952omplex(R2,Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,re(R2))),im(Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_2526_real__eq__imaginary__iff,axiom,
    ! [Y2: complex,X: complex] :
      ( member_complex(Y2,real_V2521375963428798218omplex)
     => ( member_complex(X,real_V2521375963428798218omplex)
       => ( ( X = aa_complex_complex(times_times_complex(imaginary_unit),Y2) )
        <=> ( ( X = zero_zero_complex )
            & ( Y2 = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_2527_imaginary__eq__real__iff,axiom,
    ! [Y2: complex,X: complex] :
      ( member_complex(Y2,real_V2521375963428798218omplex)
     => ( member_complex(X,real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y2) = X )
        <=> ( ( X = zero_zero_complex )
            & ( Y2 = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_2528_distinct__upto,axiom,
    ! [I2: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I2,J))) ).

% distinct_upto
tff(fact_2529_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( re(divide1717551699836669952omplex(R2,Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(re(R2)),re(Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_2530_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_2531_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),cnj(Z)) = real_V4546457046886955230omplex(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_2532_complex__cnj__mult,axiom,
    ! [X: complex,Y2: complex] : cnj(aa_complex_complex(times_times_complex(X),Y2)) = aa_complex_complex(times_times_complex(cnj(X)),cnj(Y2)) ).

% complex_cnj_mult
tff(fact_2533_complex__cnj__one,axiom,
    cnj(one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_2534_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_2535_complex__cnj__power,axiom,
    ! [X: complex,N: nat] : cnj(power_power_complex(X,N)) = power_power_complex(cnj(X),N) ).

% complex_cnj_power
tff(fact_2536_complex__cnj__add,axiom,
    ! [X: complex,Y2: complex] : cnj(plus_plus_complex(X,Y2)) = plus_plus_complex(cnj(X),cnj(Y2)) ).

% complex_cnj_add
tff(fact_2537_complex__cnj__numeral,axiom,
    ! [W: num] : cnj(numera6690914467698888265omplex(W)) = numera6690914467698888265omplex(W) ).

% complex_cnj_numeral
tff(fact_2538_complex__cnj__neg__numeral,axiom,
    ! [W: num] : cnj(uminus1482373934393186551omplex(numera6690914467698888265omplex(W))) = uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) ).

% complex_cnj_neg_numeral
tff(fact_2539_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : im(aa_complex_complex(times_times_complex(Z),cnj(Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_2540_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( re(divide1717551699836669952omplex(A,B)) = zero_zero_real )
    <=> ( re(aa_complex_complex(times_times_complex(A),cnj(B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_2541_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( im(divide1717551699836669952omplex(A,B)) = zero_zero_real )
    <=> ( im(aa_complex_complex(times_times_complex(A),cnj(B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_2542_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : real_V1022390504157884413omplex(Z) = aa_real_real(sqrt,re(aa_complex_complex(times_times_complex(Z),cnj(Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_2543_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,re(divide1717551699836669952omplex(A,B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,re(aa_complex_complex(times_times_complex(A),cnj(B)))),zero_zero_real)) ) ).

% Re_complex_div_lt_0
tff(fact_2544_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(divide1717551699836669952omplex(A,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(aa_complex_complex(times_times_complex(A),cnj(B))))) ) ).

% Re_complex_div_gt_0
tff(fact_2545_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),re(divide1717551699836669952omplex(A,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),re(aa_complex_complex(times_times_complex(A),cnj(B))))) ) ).

% Re_complex_div_ge_0
tff(fact_2546_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,re(divide1717551699836669952omplex(A,B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,re(aa_complex_complex(times_times_complex(A),cnj(B)))),zero_zero_real)) ) ).

% Re_complex_div_le_0
tff(fact_2547_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,im(divide1717551699836669952omplex(A,B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,im(aa_complex_complex(times_times_complex(A),cnj(B)))),zero_zero_real)) ) ).

% Im_complex_div_lt_0
tff(fact_2548_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),im(divide1717551699836669952omplex(A,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),im(aa_complex_complex(times_times_complex(A),cnj(B))))) ) ).

% Im_complex_div_gt_0
tff(fact_2549_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),im(divide1717551699836669952omplex(A,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),im(aa_complex_complex(times_times_complex(A),cnj(B))))) ) ).

% Im_complex_div_ge_0
tff(fact_2550_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,im(divide1717551699836669952omplex(A,B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,im(aa_complex_complex(times_times_complex(A),cnj(B)))),zero_zero_real)) ) ).

% Im_complex_div_le_0
tff(fact_2551_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(Z),cnj(Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_2552_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(divide1717551699836669952omplex(A,B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),re(aa_complex_complex(times_times_complex(A),cnj(B))))) )
      & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),im(divide1717551699836669952omplex(A,B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),im(aa_complex_complex(times_times_complex(A),cnj(B))))) ) ) ).

% complex_div_gt_0
tff(fact_2553_complex__norm__square,axiom,
    ! [Z: complex] : real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),cnj(Z)) ).

% complex_norm_square
tff(fact_2554_complex__add__cnj,axiom,
    ! [Z: complex] : plus_plus_complex(Z,cnj(Z)) = real_V4546457046886955230omplex(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),re(Z))) ).

% complex_add_cnj
tff(fact_2555_complex__diff__cnj,axiom,
    ! [Z: complex] : minus_minus_complex(Z,cnj(Z)) = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),im(Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_2556_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : divide1717551699836669952omplex(A,B) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),cnj(B)),real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(B)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_2557_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : plus_plus_complex(aa_complex_complex(times_times_complex(Z),cnj(W)),aa_complex_complex(times_times_complex(cnj(Z)),W)) = real_V4546457046886955230omplex(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),re(aa_complex_complex(times_times_complex(Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_2558_integer__of__num_I3_J,axiom,
    ! [N: num] : aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_num_Code_integer(code_integer_of_num,N),aa_num_Code_integer(code_integer_of_num,N)),one_one_Code_integer) ).

% integer_of_num(3)
tff(fact_2559_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc5135875973608410785r_bool(divide6298287555418463151nteger(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_bool_bool(fNot,dvd_dvd_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),K))) ).

% bit_cut_integer_def
tff(fact_2560_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_2561_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_2562_integer__of__num_I2_J,axiom,
    ! [N: num] : aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(aa_num_Code_integer(code_integer_of_num,N),aa_num_Code_integer(code_integer_of_num,N)) ).

% integer_of_num(2)
tff(fact_2563_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_2564_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_cp(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_2565_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_cq),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_2566_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_cr),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_2567_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_2568_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),aTP_Lamm_cs(N)),M) ).

% max_Suc2
tff(fact_2569_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),aTP_Lamm_ct(N)),M) ).

% max_Suc1
tff(fact_2570_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_al),aa_nat_nat(minus_minus_nat(M),N)) ).

% diff_Suc
tff(fact_2571_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(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,L))
           => ( ( pp(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_cu(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(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(ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_cv(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_2572_floor__real__def,axiom,
    ! [X: real] : archim6058952711729229775r_real(X) = the_int(aTP_Lamm_cw(X)) ).

% floor_real_def
tff(fact_2573_floor__rat__def,axiom,
    ! [X: rat] : archim3151403230148437115or_rat(X) = the_int(aTP_Lamm_cx(X)) ).

% floor_rat_def
tff(fact_2574_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( sgn_sgn_rat(A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( ord_less_rat(zero_zero_rat,A)
           => ( sgn_sgn_rat(A) = one_one_rat ) )
          & ( ~ ord_less_rat(zero_zero_rat,A)
           => ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_2575_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ord_less_rat(zero_zero_rat,R2)
     => ~ ! [S2: rat] :
            ( ord_less_rat(zero_zero_rat,S2)
           => ! [T2: rat] :
                ( ord_less_rat(zero_zero_rat,T2)
               => ( R2 != plus_plus_rat(S2,T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_2576_less__eq__rat__def,axiom,
    ! [X: rat,Y2: rat] :
      ( ord_less_eq_rat(X,Y2)
    <=> ( ord_less_rat(X,Y2)
        | ( X = Y2 ) ) ) ).

% less_eq_rat_def
tff(fact_2577_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_al),Nat) ).

% pred_def
tff(fact_2578_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(inverse_inverse_rat(P2)) = produc4245557441103728435nt_int(aTP_Lamm_cy,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_2579_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_2580_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% quotient_of_number(3)
tff(fact_2581_rat__one__code,axiom,
    quotient_of(one_one_rat) = product_Pair_int_int(one_one_int,one_one_int) ).

% rat_one_code
tff(fact_2582_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_2583_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = 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_2584_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int),one_one_int) ).

% quotient_of_number(4)
tff(fact_2585_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : minus_minus_rat(Q2,R2) = plus_plus_rat(Q2,uminus_uminus_rat(R2)) ).

% diff_rat_def
tff(fact_2586_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : divide_divide_rat(Q2,R2) = times_times_rat(Q2,inverse_inverse_rat(R2)) ).

% divide_rat_def
tff(fact_2587_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ord_less_rat(P2,Q2)
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_da(Q2)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_2588_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ord_less_eq_rat(P2,Q2)
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_dc(Q2)),quotient_of(P2))) ) ).

% rat_less_eq_code
tff(fact_2589_quotient__of__int,axiom,
    ! [A: int] : quotient_of(of_int(A)) = product_Pair_int_int(A,one_one_int) ).

% quotient_of_int
tff(fact_2590_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(plus_plus_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_de(Q2),quotient_of(P2)) ).

% rat_plus_code
tff(fact_2591_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(minus_minus_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_dg(Q2),quotient_of(P2)) ).

% rat_minus_code
tff(fact_2592_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(product_Pair_int_int(P2,zero_zero_int)) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% normalize_denom_zero
tff(fact_2593_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P2,Q2)) = normalize(product_Pair_int_int(R2,S)) )
         => ( times_times_int(P2,S) = times_times_int(R2,Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_2594_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(times_times_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_di(Q2),quotient_of(P2)) ).

% rat_times_code
tff(fact_2595_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(divide_divide_rat(P2,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_dk(Q2),quotient_of(P2)) ).

% rat_divide_code
tff(fact_2596_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_2597_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_2598_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y2: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y2 )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(minus_minus_nat(X),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y2 = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(minus_minus_nat(Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_2599_Frct__code__post_I3_J,axiom,
    frct(product_Pair_int_int(one_one_int,one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_2600_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_2601_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(minus_minus_nat(K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_2602_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y2: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y2 )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(minus_minus_nat(X),Xa2)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y2 = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(minus_minus_nat(Xa2),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_2603_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(K)) = product_fst_nat_nat(unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_2604_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se8568078237143864401it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = bit_se8568078237143864401it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_2605_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),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_2606_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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_2607_drop__bit__minus__one,axiom,
    ! [N: nat] : 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_2608_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] : product_fst_nat_nat(divmod_nat(M,N)) = aa_nat_nat(divide_divide_nat(M),N) ).

% fst_divmod_nat
tff(fact_2609_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : 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)))) = bit_se8568078237143864401it_int(N,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_2610_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se8568078237143864401it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = bit_se8568078237143864401it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_2611_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : 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)))) = bit_se8568078237143864401it_int(N,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_2612_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : bit_se8568078237143864401it_int(M,bit_se545348938243370406it_int(N,K)) = bit_se8568078237143864401it_int(aa_nat_nat(minus_minus_nat(M),N),bit_se545348938243370406it_int(aa_nat_nat(minus_minus_nat(N),M),K)) ).

% drop_bit_push_bit_int
tff(fact_2613_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se8568078237143864401it_int(N,K) = divide_divide_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% drop_bit_int_def
tff(fact_2614_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(K)) = product_snd_nat_nat(unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_2615_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( finite_finite_nat(S3)
     => ? [R3: fun_nat_nat] :
          ( strict1292158309912662752at_nat(R3,set_ord_lessThan_nat(finite_card_nat(S3)))
          & ! [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N6),finite_card_nat(S3)))
             => member_nat(aa_nat_nat(R3,N6),S3) ) ) ) ).

% finite_enumerate
tff(fact_2616_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(bit_se8570568707652914677it_nat(N),aa_nat_nat(suc,zero_zero_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_2617_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(bit_se8570568707652914677it_nat(N),nat2(K)) = nat2(bit_se8568078237143864401it_int(N,K)) ).

% drop_bit_nat_eq
tff(fact_2618_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(bit_se8570568707652914677it_nat(N),M) = aa_nat_nat(divide_divide_nat(M),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).

% drop_bit_nat_def
tff(fact_2619_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(sgn_sgn_rat(P2)) = product_Pair_int_int(sgn_sgn_int(product_fst_int_int(quotient_of(P2))),one_one_int) ).

% rat_sgn_code
tff(fact_2620_bezw__non__0,axiom,
    ! [Y2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y2))
     => ( bezw(X,Y2) = product_Pair_int_int(product_snd_int_int(bezw(Y2,aa_nat_nat(modulo_modulo_nat(X),Y2))),minus_minus_int(product_fst_int_int(bezw(Y2,aa_nat_nat(modulo_modulo_nat(X),Y2))),times_times_int(product_snd_int_int(bezw(Y2,aa_nat_nat(modulo_modulo_nat(X),Y2))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Y2))))) ) ) ).

% bezw_non_0
tff(fact_2621_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y2: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y2 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y2 = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y2 = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),minus_minus_int(product_fst_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),times_times_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_2622_bezw_Osimps,axiom,
    ! [Y2: nat,X: nat] :
      ( ( ( Y2 = zero_zero_nat )
       => ( bezw(X,Y2) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y2 != zero_zero_nat )
       => ( bezw(X,Y2) = product_Pair_int_int(product_snd_int_int(bezw(Y2,aa_nat_nat(modulo_modulo_nat(X),Y2))),minus_minus_int(product_fst_int_int(bezw(Y2,aa_nat_nat(modulo_modulo_nat(X),Y2))),times_times_int(product_snd_int_int(bezw(Y2,aa_nat_nat(modulo_modulo_nat(X),Y2))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Y2))))) ) ) ) ).

% bezw.simps
tff(fact_2623_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y2: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y2 )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y2 = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y2 = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),minus_minus_int(product_fst_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),times_times_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Xa2))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_2624_one__mod__minus__numeral,axiom,
    ! [N: num] : 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,adjust_mod(aa_num_int(numeral_numeral_int,N),product_snd_int_int(unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_2625_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = adjust_mod(aa_num_int(numeral_numeral_int,N),product_snd_int_int(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_mod_numeral
tff(fact_2626_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : 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,adjust_mod(aa_num_int(numeral_numeral_int,N),product_snd_int_int(unique5052692396658037445od_int(M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_2627_minus__one__mod__numeral,axiom,
    ! [N: num] : modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = adjust_mod(aa_num_int(numeral_numeral_int,N),product_snd_int_int(unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_2628_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P2)))
       => ( normalize(P2) = product_Pair_int_int(divide_divide_int(product_fst_int_int(P2),gcd_gcd_int(product_fst_int_int(P2),product_snd_int_int(P2))),divide_divide_int(product_snd_int_int(P2),gcd_gcd_int(product_fst_int_int(P2),product_snd_int_int(P2)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P2)))
       => ( ( ( product_snd_int_int(P2) = zero_zero_int )
           => ( normalize(P2) = product_Pair_int_int(zero_zero_int,one_one_int) ) )
          & ( ( product_snd_int_int(P2) != zero_zero_int )
           => ( normalize(P2) = product_Pair_int_int(divide_divide_int(product_fst_int_int(P2),aa_int_int(uminus_uminus_int,gcd_gcd_int(product_fst_int_int(P2),product_snd_int_int(P2)))),divide_divide_int(product_snd_int_int(P2),aa_int_int(uminus_uminus_int,gcd_gcd_int(product_fst_int_int(P2),product_snd_int_int(P2))))) ) ) ) ) ) ).

% normalize_def
tff(fact_2629_gcd__1__int,axiom,
    ! [M: int] : gcd_gcd_int(M,one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_2630_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : gcd_gcd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)),X) = gcd_gcd_int(aa_num_int(numeral_numeral_int,N),X) ).

% gcd_neg_numeral_1_int
tff(fact_2631_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : gcd_gcd_int(X,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = gcd_gcd_int(X,aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2_int
tff(fact_2632_gcd__ge__0__int,axiom,
    ! [X: int,Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_gcd_int(X,Y2))) ).

% gcd_ge_0_int
tff(fact_2633_bezout__int,axiom,
    ! [X: int,Y2: int] :
    ? [U2: int,V2: int] : plus_plus_int(times_times_int(U2,X),times_times_int(V2,Y2)) = gcd_gcd_int(X,Y2) ).

% bezout_int
tff(fact_2634_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] : times_times_int(aa_int_int(abs_abs_int,K),gcd_gcd_int(M,N)) = gcd_gcd_int(times_times_int(K,M),times_times_int(K,N)) ).

% gcd_mult_distrib_int
tff(fact_2635_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,gcd_gcd_int(A,B)),A)) ) ).

% gcd_le1_int
tff(fact_2636_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,gcd_gcd_int(A,B)),B)) ) ).

% gcd_le2_int
tff(fact_2637_gcd__cases__int,axiom,
    ! [X: int,Y2: 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),Y2))
         => pp(aa_int_bool(P,gcd_gcd_int(X,Y2))) ) )
     => ( ( 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,Y2),zero_zero_int))
           => pp(aa_int_bool(P,gcd_gcd_int(X,aa_int_int(uminus_uminus_int,Y2)))) ) )
       => ( ( 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),Y2))
             => pp(aa_int_bool(P,gcd_gcd_int(aa_int_int(uminus_uminus_int,X),Y2))) ) )
         => ( ( 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,Y2),zero_zero_int))
               => pp(aa_int_bool(P,gcd_gcd_int(aa_int_int(uminus_uminus_int,X),aa_int_int(uminus_uminus_int,Y2)))) ) )
           => pp(aa_int_bool(P,gcd_gcd_int(X,Y2))) ) ) ) ) ).

% gcd_cases_int
tff(fact_2638_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 = gcd_gcd_int(A,B) ) ) ).

% gcd_unique_int
tff(fact_2639_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(gcd_gcd_nat(M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_2640_gcd__Suc__0,axiom,
    ! [M: nat] : aa_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_2641_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(gcd_gcd_nat(M),N)))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_2642_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(gcd_gcd_nat(M),N)) = aa_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_2643_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(gcd_gcd_nat(A),B)),B)) ) ).

% gcd_le2_nat
tff(fact_2644_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(gcd_gcd_nat(A),B)),A)) ) ).

% gcd_le1_nat
tff(fact_2645_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(gcd_gcd_nat(aa_nat_nat(minus_minus_nat(N),M)),N) = aa_nat_nat(gcd_gcd_nat(M),N) ) ) ).

% gcd_diff2_nat
tff(fact_2646_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(gcd_gcd_nat(aa_nat_nat(minus_minus_nat(M),N)),N) = aa_nat_nat(gcd_gcd_nat(M),N) ) ) ).

% gcd_diff1_nat
tff(fact_2647_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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(gcd_gcd_nat(A),B)) ) ).

% bezout_nat
tff(fact_2648_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(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(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(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(gcd_gcd_nat(A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_2649_bezw__aux,axiom,
    ! [X: nat,Y2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(gcd_gcd_nat(X),Y2)) = plus_plus_int(times_times_int(product_fst_int_int(bezw(X,Y2)),aa_nat_int(semiri1314217659103216013at_int,X)),times_times_int(product_snd_int_int(bezw(X,Y2)),aa_nat_int(semiri1314217659103216013at_int,Y2))) ).

% bezw_aux
tff(fact_2650_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))
           => ( ! [I: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I))
                 => pp(aa_nat_bool(P,I)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_2651_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = nat2(minus_minus_int(U,plus_plus_int(L,one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_2652_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] : bit_se6526347334894502574or_int(K,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = bit_ri7919022796975470100ot_int(bit_se6526347334894502574or_int(K,neg_numeral_sub_int(N,one))) ).

% xor_minus_numerals(2)
tff(fact_2653_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] : bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)),K) = bit_ri7919022796975470100ot_int(bit_se6526347334894502574or_int(neg_numeral_sub_int(N,one),K)) ).

% xor_minus_numerals(1)
tff(fact_2654_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(plus_plus_int(L,one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_2655_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub_int(bitM(N),one) = times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),neg_numeral_sub_int(N,one)) ).

% sub_BitM_One_eq
tff(fact_2656_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or5832277885323065728an_int(I2,J) = set_int2(upto(plus_plus_int(I2,one_one_int),minus_minus_int(J,one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_2657_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(set_or5834768355832116004an_nat(L,U)) = aa_nat_nat(minus_minus_nat(U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_2658_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_2659_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_2660_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(ord_le3102999989581377725nteger(K,zero_z3403309356797280102nteger))
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_2661_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = plus_plus_nat(N) ).

% Suc_funpow
tff(fact_2662_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = numeral_numeral_nat(K) ).

% nat_of_integer_code_post(3)
tff(fact_2663_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_2664_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(ord_le3102999989581377725nteger(K,zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = zero_zero_nat ) )
      & ( ~ pp(ord_le3102999989581377725nteger(K,zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_dl,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_2665_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(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_dm(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_2666_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_cg,aTP_Lamm_ak) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_2667_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger))
       => ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_dn,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_2668_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_2669_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(X,Xa2)) = plus_plus_int(code_int_of_integer(X),code_int_of_integer(Xa2)) ).

% plus_integer.rep_eq
tff(fact_2670_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa2)) = times_times_int(code_int_of_integer(X),code_int_of_integer(Xa2)) ).

% times_integer.rep_eq
tff(fact_2671_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_2672_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_2673_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(ord_le3102999989581377725nteger(X,Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).

% less_eq_integer.rep_eq
tff(fact_2674_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(ord_le3102999989581377725nteger(K,L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_eq_iff
tff(fact_2675_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_2676_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_2677_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : times_times_int(abs_Integ(Xa2),abs_Integ(X)) = abs_Integ(aa_Pro262675715343573244at_nat(produc4528391901023682385at_nat(aTP_Lamm_dp,Xa2),X)) ).

% times_int.abs_eq
tff(fact_2678_bij__betw__Suc,axiom,
    ! [M7: set_nat,N2: set_nat] :
      ( bij_betw_nat_nat(suc,M7,N2)
    <=> ( image_nat_nat(suc,M7) = N2 ) ) ).

% bij_betw_Suc
tff(fact_2679_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I2,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I2),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_2680_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I2,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I2),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_2681_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ member_nat(zero_zero_nat,image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_2682_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,set_ord_lessThan_nat(N)) = set_or1269000886237332187st_nat(one_one_nat,N) ).

% image_Suc_lessThan
tff(fact_2683_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,set_ord_atMost_nat(N)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_2684_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_2685_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_2686_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_lessThan_nat(N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_2687_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_atMost_nat(aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_atMost_nat(N))) ).

% atMost_Suc_eq_insert_0
tff(fact_2688_one__int__def,axiom,
    one_one_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_2689_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,abs_Integ(Xa2)),abs_Integ(X)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc5023667602222594023t_bool(aTP_Lamm_dr,Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_2690_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,abs_Integ(Xa2)),abs_Integ(X)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc5023667602222594023t_bool(aTP_Lamm_dt,Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_2691_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : plus_plus_int(abs_Integ(Xa2),abs_Integ(X)) = abs_Integ(aa_Pro262675715343573244at_nat(produc4528391901023682385at_nat(aTP_Lamm_dv,Xa2),X)) ).

% plus_int.abs_eq
tff(fact_2692_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : minus_minus_int(abs_Integ(Xa2),abs_Integ(X)) = abs_Integ(aa_Pro262675715343573244at_nat(produc4528391901023682385at_nat(aTP_Lamm_dx,Xa2),X)) ).

% minus_int.abs_eq
tff(fact_2693_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,set_ord_lessThan_nat(nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_2694_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y2: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y2))
       => ( image_nat_nat(aTP_Lamm_dy(C),set_or4665077453230672383an_nat(X,Y2)) = set_or4665077453230672383an_nat(aa_nat_nat(minus_minus_nat(X),C),aa_nat_nat(minus_minus_nat(Y2),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y2))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2))
           => ( image_nat_nat(aTP_Lamm_dy(C),set_or4665077453230672383an_nat(X,Y2)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y2))
           => ( image_nat_nat(aTP_Lamm_dy(C),set_or4665077453230672383an_nat(X,Y2)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_2695_suminf__eq__SUP__real,axiom,
    ! [X6: fun_nat_real] :
      ( summable_real(X6)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X6,I3)))
       => ( suminf_real(X6) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_dz(X6),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_2696_Inf__real__def,axiom,
    ! [X6: set_real] : comple4887499456419720421f_real(X6) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X6))) ).

% Inf_real_def
tff(fact_2697_finite__int__iff__bounded__le,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
    <=> ? [K3: int] : ord_less_eq_set_int(image_int_int(abs_abs_int,S3),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_2698_finite__int__iff__bounded,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
    <=> ? [K3: int] : ord_less_eq_set_int(image_int_int(abs_abs_int,S3),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_2699_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_ea(L),set_or4662586982721622107an_int(zero_zero_int,minus_minus_int(U,L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_2700_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aTP_Lamm_eb(N),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N) ) ) ).

% range_mod
tff(fact_2701_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_2702_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_2703_card__UNIV__bool,axiom,
    finite_card_bool(top_top_set_bool) = numeral_numeral_nat(aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_2704_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(times_times_real(A),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
      & ( ( A != zero_zero_real )
       => ( image_real_real(times_times_real(A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_2705_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_ec,N)),X) ) ) ) ).

% root_def
tff(fact_2706_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_2707_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_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_2708_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_2709_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_2710_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( set_or4665077453230672383an_nat(I2,aa_nat_nat(plus_plus_nat(J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I2,J),set_or4665077453230672383an_nat(J,aa_nat_nat(plus_plus_nat(J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_2711_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)),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_2712_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_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_2713_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B7: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B7)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(B7)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B62))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B52))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B42))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B32))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B22))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B1))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger(B0)) ).

% integer_of_char_code
tff(fact_2714_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(bit_se2925701944663578781it_nat(numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_2715_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,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(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_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_2716_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H2))
               => ( member_real(aa_real_real(plus_plus_real(X),H2),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H2),D2))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(plus_plus_real(X),H2))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_2717_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H2))
               => ( member_real(aa_real_real(plus_plus_real(X),H2),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H2),D2))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(plus_plus_real(X),H2)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_2718_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H2),D2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(plus_plus_real(X),H2))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_2719_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [H2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H2),D2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(plus_plus_real(X),H2)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_2720_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ( ! [X3: real] :
            ( member_real(X3,set_or1222579329274155063t_real(A,B))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X3))) )
       => ( 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(G,A)),aa_real_real(G,B))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_2721_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
             => ? [Y: real] :
                  ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(X3,top_top_set_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(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_2722_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
             => ? [Y: real] :
                  ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(X3,top_top_set_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,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_2723_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
             => ? [Y: real] :
                  ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(X3,top_top_set_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,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_2724_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
             => ? [Y: real] :
                  ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(X3,top_top_set_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(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_2725_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_2726_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F2: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
            & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),aa_real_real(F2,Z3)) ) ) ) ) ).

% MVT2
tff(fact_2727_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_2728_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_2729_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,X))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_2730_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_2731_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ed,N),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_2732_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_ee(G),N),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X)),aa_nat_nat(minus_minus_nat(N),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_2733_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_ef(R2),aa_real_real(times_times_real(R2),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_2734_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(log(B),aa_real_real(divide_divide_real(one_one_real),aa_real_real(times_times_real(aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_2735_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_eg(G),R2),aa_real_real(times_times_real(aa_real_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_2736_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_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(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_eh(G),F),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(R2),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(M),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_2737_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))),numeral_numeral_real(aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_2738_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F2: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L4: fun_nat_real] :
      ( ! [N3: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ei(F),N3),aa_nat_real(aa_real_fun_nat_real(F2,X0),N3),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X3)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F2,X0))
           => ( summable_real(L4)
             => ( ! [N3: nat,X3: real,Y3: real] :
                    ( member_real(X3,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y3,set_or1633881224788618240n_real(A,B))
                     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X3),N3)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N3)))),aa_real_real(times_times_real(aa_nat_real(L4,N3)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),Y3))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_ej(F),suminf_real(aa_real_fun_nat_real(F2,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_2739_DERIV__arctan,axiom,
    ! [X: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_2740_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_2741_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D5: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D5 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D5 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_2742_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_2743_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_2744_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_ek(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_em(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ek(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_2745_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(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_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_2746_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_2747_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(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_2748_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 )
     => ( ! [M5: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_en(Diff),X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_2749_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 )
        & ! [M5: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_en(Diff),X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_2750_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_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_2751_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 )
         => ( ! [M5: nat,T2: 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),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_eo(H),Diff),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_2752_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 )
       => ( ! [M5: nat,T2: 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),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_eo(H),Diff),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_2753_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 )
         => ( ! [M5: nat,T2: 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,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,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_eo(H),Diff),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_2754_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 )
         => ( ! [M5: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_en(Diff),X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_2755_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 )
     => ( ! [M5: nat,T2: 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,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_en(Diff),X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_2756_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 )
       => ( ! [M5: nat,T2: 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,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ep(Diff),C),X),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_2757_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 )
       => ( ! [M5: nat,T2: 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,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_eq(Diff),B),C),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_2758_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 )
       => ( ! [M5: nat,T2: 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,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),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(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_eq(Diff),A),C),set_ord_lessThan_nat(N))),aa_real_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(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_2759_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M5: nat,T2: 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),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M3: nat,T3: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),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_es(N),Diff),B2),M3),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3)),aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_et(Diff),M3),T3),set_ord_lessThan_nat(aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,M3))))),aa_real_real(times_times_real(B2),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T3),aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,M3)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,M3))))))),topolo2177554685111907308n_real(T3,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_2760_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_eu,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ev,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_2761_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D5: 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,numeral_numeral_nat(aa_num_num(bit0,one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_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,numeral_numeral_nat(aa_num_num(bit0,one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D5 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_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,numeral_numeral_nat(aa_num_num(bit0,one))),N))
               => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(minus_minus_nat(N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_2762_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X3: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ? [L5: real,M9: real] :
            ( ! [X2: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B)) )
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L5),aa_real_real(F,X2)))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),M9)) ) )
            & ! [Y: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L5),Y))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),M9)) )
               => ? [X3: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
                    & ( aa_real_real(F,X3) = Y ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_2763_isCont__real__sqrt,axiom,
    ! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_2764_isCont__real__root,axiom,
    ! [X: real,N: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),aa_nat_fun_real_real(root,N)) ).

% isCont_real_root
tff(fact_2765_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_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))
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
               => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_2766_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_2767_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_ew,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_2768_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_2769_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_2770_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(B,X))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).

% LIM_less_bound
tff(fact_2771_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_2772_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z3),X))),D))
           => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z3),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_2773_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [Z3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C2)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F2,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_2774_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_ex(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ex(A),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N6))),groups6591440286371151544t_real(aTP_Lamm_ex(A),set_ord_lessThan_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N6)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_2775_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_ex(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ex(A),set_ord_lessThan_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N6)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_ex(A),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N6))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_2776_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_2777_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_2778_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(aTP_Lamm_ey(C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_2779_monoseq__convergent,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( topolo6980174941875973593q_real(X6)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X6,I3))),B2))
       => ~ ! [L5: real] : ~ filterlim_nat_real(X6,topolo2815343760600316023s_real(L5),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_2780_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_ez,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_2781_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N3))),aa_nat_real(G,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(G,N3)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(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_2782_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N5: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N3))) )
     => filterlim_nat_real(aTP_Lamm_fb(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_2783_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_fc,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_2784_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fd,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_2785_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_fe,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_2786_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ff,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_2787_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),L))
       => ( ! [E2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
             => ? [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(plus_plus_real(aa_nat_real(F,N6)),E2))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_2788_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(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_2789_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_fg(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_2790_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(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_2791_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(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_2792_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: 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_fh,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_2793_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fi,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_2794_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fj,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_2795_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fk,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_2796_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_ex(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_2797_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => summable_real(aTP_Lamm_ex(A)) ) ) ) ).

% summable
tff(fact_2798_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fl(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_fm(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_2799_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_fn(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_fm(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_2800_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_fo(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ex(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_2801_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(aa_real_fun_nat_real(aTP_Lamm_av,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_2802_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => filterlim_nat_real(aTP_Lamm_fo(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ex(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_2803_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ex(A),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_ex(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_2804_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( 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_ex(A),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N6)))),L2))
              & filterlim_nat_real(aTP_Lamm_fo(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_ex(A),set_ord_lessThan_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N6)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_fp(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_2805_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_fp(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ex(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_2806_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => filterlim_nat_real(aTP_Lamm_fp(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ex(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_2807_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_ex(A))),groups6591440286371151544t_real(aTP_Lamm_ex(A),set_ord_lessThan_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_2808_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C3: real] :
        ! [X4: real] : aa_real_real(F,X4) = aa_real_real(times_times_real(X4),C3) ) ).

% real_bounded_linear
tff(fact_2809_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_fq(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_2810_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_2811_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),numeral_numeral_real(aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_2812_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_2813_greaterThan__0,axiom,
    set_or1210151606488870762an_nat(zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_2814_greaterThan__Suc,axiom,
    ! [K: nat] : set_or1210151606488870762an_nat(aa_nat_nat(suc,K)) = minus_minus_set_nat(set_or1210151606488870762an_nat(K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_2815_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_2816_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_2817_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
         => ? [Y: real] :
              ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(X3,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_2818_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F3: 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,F3)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fr(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_2819_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F3: 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,F3)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fr(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_2820_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_2821_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_2822_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_2823_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_2824_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_2825_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_fs,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_2826_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_ft(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_2827_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_2828_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X3))
         => ? [Y: real] :
              ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(X3,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_2829_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_2830_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X3: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ( ! [X3: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_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_eq_real,X3),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
           => ( ! [X3: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_2831_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_fu(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_2832_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_fv(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_2833_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_fv(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_2834_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N7: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),F3) ) ).

% le_sequentially
tff(fact_2835_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N7: nat] :
        ! [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N4))
         => pp(aa_nat_bool(P,N4)) ) ) ).

% eventually_sequentially
tff(fact_2836_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X3))
         => pp(aa_nat_bool(P,X3)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_2837_eventually__at__right__to__0,axiom,
    ! [P: fun_real_bool,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fw(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_2838_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A: real,B: real] :
      ( ord_less_eq_set_real(image_nat_real(F,top_top_set_nat),set_or1222579329274155063t_real(A,B))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_2839_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(aa_real_fun_nat_real(power_power_real,X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_2840_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B: nat] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_2841_Greatest__le__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)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_2842_GreatestI__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)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_2843_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_2844_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K)) = set_or1210151606488870762an_nat(K) ).

% atLeast_Suc_greaterThan
tff(fact_2845_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(plus_plus_int(L,one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_2846_decseq__bounded,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X6)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X6,I3)))
       => bfun_nat_real(X6,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_2847_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or6656581121297822940st_int(I2,J) = set_int2(upto(plus_plus_int(I2,one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_2848_decseq__convergent,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_9091379641038594480t_real(X6)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X6,I3)))
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X6,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L5),aa_nat_real(X6,I))) ) ) ) ).

% decseq_convergent
tff(fact_2849_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K)) = minus_minus_set_nat(set_ord_atLeast_nat(K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_2850_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ? [L2: real,Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
              & has_fi5821293074295781190e_real(F,L2,topolo2177554685111907308n_real(Z3,top_top_set_real))
              & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),L2) ) ) ) ) ) ).

% MVT
tff(fact_2851_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_fx(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_2852_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ? [C2: real,D2: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C2,D2) )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_2853_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_2854_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_2855_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_2856_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_fy(F)) ) ) ).

% continuous_on_artanh'
tff(fact_2857_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_2858_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( 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))
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_2859_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(produc5023667602222594023t_bool(aTP_Lamm_dt,rep_Integ(X)),rep_Integ(Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_2860_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(produc5023667602222594023t_bool(aTP_Lamm_dr,rep_Integ(X)),rep_Integ(Xa2))) ) ).

% less_int.rep_eq
tff(fact_2861_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_2862_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_2863_incseq__bounded,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X6)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,I3)),B2))
       => bfun_nat_real(X6,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_2864_incseq__convergent,axiom,
    ! [X6: fun_nat_real,B2: real] :
      ( order_mono_nat_real(X6)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,I3)),B2))
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X6,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,I)),L5)) ) ) ) ).

% incseq_convergent
tff(fact_2865_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K))
     => order_mono_nat_nat(aTP_Lamm_fz(K)) ) ).

% mono_ge2_power_minus_self
tff(fact_2866_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_ec,N),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_2867_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => inj_on_real_real(log(B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_2868_inj__on__diff__nat,axiom,
    ! [N2: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( member_nat(N3,N2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N3)) )
     => inj_on_nat_nat(aTP_Lamm_dy(K),N2) ) ).

% inj_on_diff_nat
tff(fact_2869_inj__Suc,axiom,
    ! [N2: set_nat] : inj_on_nat_nat(suc,N2) ).

% inj_Suc
tff(fact_2870_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( inj_on_nat_nat(G,top_top_set_nat)
       => ( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_2871_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( inj_on_nat_nat(G,top_top_set_nat)
       => ( ! [X3: nat] : 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,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F))) ) ) ) ).

% suminf_reindex_mono
tff(fact_2872_inj__on__char__of__nat,axiom,
    inj_on_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_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_2873_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( inj_on_nat_nat(G,top_top_set_nat)
       => ( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
         => ( ! [X3: nat] :
                ( ~ member_nat(X3,image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X3) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_2874_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_ga(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_2875_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_2876_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),ring_1_of_int_real(N)) = power_int_real(X,N) ) ) ) ).

% powr_real_of_int'
tff(fact_2877_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))
       => ( num_of_nat(aa_nat_nat(suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( num_of_nat(aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_2878_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(numeral_numeral_nat(Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_2879_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_2880_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( numeral_numeral_nat(num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_2881_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( num_of_nat(N) = one ) ) ).

% num_of_nat_One
tff(fact_2882_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( num_of_nat(aa_nat_nat(plus_plus_nat(N),N)) = aa_num_num(bit0,num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_2883_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))
       => ( num_of_nat(aa_nat_nat(plus_plus_nat(M),N)) = plus_plus_num(num_of_nat(M),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_2884_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_gb)) ).

% pred_nat_def
tff(fact_2885_pow_Osimps_I3_J,axiom,
    ! [X: num,Y2: num] : pow(X,aa_num_num(bit1,Y2)) = times_times_num(sqr(pow(X,Y2)),X) ).

% pow.simps(3)
tff(fact_2886_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_lessThan_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_lessThan_nat(K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_2887_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_atMost_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_atMost_nat(K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_2888_sqr_Osimps_I2_J,axiom,
    ! [N: num] : sqr(aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,sqr(N))) ).

% sqr.simps(2)
tff(fact_2889_sqr_Osimps_I1_J,axiom,
    sqr(one) = one ).

% sqr.simps(1)
tff(fact_2890_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = times_times_num(X,X) ).

% sqr_conv_mult
tff(fact_2891_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I2)),J))
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_2892_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),J))
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_2893_pow_Osimps_I2_J,axiom,
    ! [X: num,Y2: num] : pow(X,aa_num_num(bit0,Y2)) = sqr(pow(X,Y2)) ).

% pow.simps(2)
tff(fact_2894_sqr_Osimps_I3_J,axiom,
    ! [N: num] : sqr(aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,plus_plus_num(sqr(N),N))) ).

% sqr.simps(3)
tff(fact_2895_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(minus_minus_nat(J),I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J))),N) = aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_2896_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(minus_minus_nat(J),aa_nat_nat(suc,I2))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J))),N) = aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_2897_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y2) )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( pp(Y2)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( pp(Y2)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),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(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_2898_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))))
                  & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),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(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_2899_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg4))
    <=> ( ( Deg = Deg4 )
        & ! [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(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(minus_minus_nat(Deg),aa_nat_nat(divide_divide_nat(Deg),numeral_numeral_nat(aa_num_num(bit0,one))))))
        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg),aa_nat_nat(divide_divide_nat(Deg),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(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg),TreeList),Summary)),Mima2) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_2900_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))))
                & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),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(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_2901_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y2: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y2) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( pp(Y2)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( pp(Y2)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),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(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg2),TreeList2),Summary2)),Mima) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_2902_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),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(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_2903_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_gd(X6)) ).

% Sup_int_def
tff(fact_2904_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),numeral_numeral_nat(aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Deg2),aa_nat_nat(divide_divide_nat(Deg2),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(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_gc(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_2905_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_2906_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi2905341329935302413cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_2907_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_2908_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_2909_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_2910_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_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,pred_numeral(K)),N)) ).

% min_numeral_Suc
tff(fact_2911_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_2912_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_2913_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(minus_minus_nat(M),N),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_2914_min__diff,axiom,
    ! [M: nat,I2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(minus_minus_nat(M),I2)),aa_nat_nat(minus_minus_nat(N),I2)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I2) ).

% min_diff
tff(fact_2915_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_2916_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = 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),Q2)) ).

% nat_mult_min_right
tff(fact_2917_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: 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),bit_se2923211474154528505it_int(aa_nat_nat(minus_minus_nat(M),N),L)) ).

% take_bit_concat_bit_eq
tff(fact_2918_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,aTP_Lamm_ge(N)),M) ).

% min_Suc2
tff(fact_2919_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,aTP_Lamm_gf(N)),M) ).

% min_Suc1
tff(fact_2920_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_gg) ).

% Rats_eq_int_div_nat
tff(fact_2921_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q2),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_2922_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_2923_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_2924_Rats__dense__in__real,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y2))
     => ? [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),Y2)) ) ) ).

% Rats_dense_in_real
tff(fact_2925_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_2926_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_2927_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_gh) ).

% Rats_eq_int_div_int
tff(fact_2928_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_2929_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_rat(ring_1_of_int_rat(divide_divide_int(A,B)),fract(A,B))
      & ord_less_rat(fract(A,B),ring_1_of_int_rat(plus_plus_int(divide_divide_int(A,B),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_2930_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : times_times_rat(fract(A,B),fract(C,D)) = fract(times_times_int(A,C),times_times_int(B,D)) ).

% mult_rat
tff(fact_2931_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : divide_divide_rat(fract(A,B),fract(C,D)) = fract(times_times_int(A,D),times_times_int(B,C)) ).

% divide_rat
tff(fact_2932_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ord_less_rat(fract(A,B),fract(C,D))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,times_times_int(times_times_int(A,D),times_times_int(B,D))),times_times_int(times_times_int(C,B),times_times_int(B,D)))) ) ) ) ).

% less_rat
tff(fact_2933_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( plus_plus_rat(fract(A,B),fract(C,D)) = fract(plus_plus_int(times_times_int(A,D),times_times_int(C,B)),times_times_int(B,D)) ) ) ) ).

% add_rat
tff(fact_2934_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ord_less_eq_rat(fract(A,B),fract(C,D))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,times_times_int(times_times_int(A,D),times_times_int(B,D))),times_times_int(times_times_int(C,B),times_times_int(B,D)))) ) ) ) ).

% le_rat
tff(fact_2935_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( minus_minus_rat(fract(A,B),fract(C,D)) = fract(minus_minus_int(times_times_int(A,D),times_times_int(C,B)),times_times_int(B,D)) ) ) ) ).

% diff_rat
tff(fact_2936_sgn__rat,axiom,
    ! [A: int,B: int] : sgn_sgn_rat(fract(A,B)) = ring_1_of_int_rat(times_times_int(sgn_sgn_int(A),sgn_sgn_int(B))) ).

% sgn_rat
tff(fact_2937_Fract__of__int__eq,axiom,
    ! [K: int] : fract(K,one_one_int) = ring_1_of_int_rat(K) ).

% Fract_of_int_eq
tff(fact_2938_One__rat__def,axiom,
    one_one_rat = fract(one_one_int,one_one_int) ).

% One_rat_def
tff(fact_2939_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( fract(A,B) = fract(C,D) )
        <=> ( times_times_int(A,D) = times_times_int(C,B) ) ) ) ) ).

% eq_rat(1)
tff(fact_2940_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( fract(times_times_int(C,A),times_times_int(C,B)) = fract(A,B) ) ) ).

% mult_rat_cancel
tff(fact_2941_eq__rat_I2_J,axiom,
    ! [A: int] : fract(A,zero_zero_int) = fract(zero_zero_int,one_one_int) ).

% eq_rat(2)
tff(fact_2942_Fract__of__nat__eq,axiom,
    ! [K: nat] : fract(aa_nat_int(semiri1314217659103216013at_int,K),one_one_int) = semiri681578069525770553at_rat(K) ).

% Fract_of_nat_eq
tff(fact_2943_Zero__rat__def,axiom,
    zero_zero_rat = fract(zero_zero_int,one_one_int) ).

% Zero_rat_def
tff(fact_2944_rat__number__collapse_I3_J,axiom,
    ! [W: num] : fract(aa_num_int(numeral_numeral_int,W),one_one_int) = numeral_numeral_rat(W) ).

% rat_number_collapse(3)
tff(fact_2945_rat__number__expand_I3_J,axiom,
    ! [K: num] : numeral_numeral_rat(K) = fract(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% rat_number_expand(3)
tff(fact_2946_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))
     => ( ord_less_rat(one_one_rat,fract(A,B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_2947_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))
     => ( ord_less_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_2948_rat__number__collapse_I5_J,axiom,
    fract(aa_int_int(uminus_uminus_int,one_one_int),one_one_int) = uminus_uminus_rat(one_one_rat) ).

% rat_number_collapse(5)
tff(fact_2949_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( fract(plus_plus_int(M,N),N) = plus_plus_rat(fract(M,N),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_2950_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))
     => ( ord_less_eq_rat(zero_zero_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_2951_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))
     => ( ord_less_eq_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_2952_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))
     => ( ord_less_eq_rat(one_one_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_2953_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))
     => ( ord_less_eq_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_2954_rat__number__collapse_I4_J,axiom,
    ! [W: num] : fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W)),one_one_int) = uminus_uminus_rat(numeral_numeral_rat(W)) ).

% rat_number_collapse(4)
tff(fact_2955_rat__number__expand_I5_J,axiom,
    ! [K: num] : uminus_uminus_rat(numeral_numeral_rat(K)) = fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% rat_number_expand(5)
tff(fact_2956_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : bit_se2923211474154528505it_int(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_gi(M),bit_take_bit_num(numeral_numeral_nat(M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_2957_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_2958_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_2959_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(numeral_numeral_nat(R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_2960_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one) = case_nat_option_num(none_num,aTP_Lamm_gj,N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_2961_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(bit_se2925701944663578781it_nat(N),numeral_numeral_nat(M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(bit_se2925701944663578781it_nat(N),numeral_numeral_nat(M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = some_num(num_of_nat(aa_nat_nat(bit_se2925701944663578781it_nat(N),numeral_numeral_nat(M)))) ) ) ) ).

% take_bit_num_def
tff(fact_2962_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : 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,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_2963_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : 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,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_2964_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_2965_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_gk,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_2966_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(numeral_numeral_nat(R2),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_2967_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(numeral_numeral_nat(R2),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_gk,bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_2968_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : 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_2969_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : 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_2970_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_gl,bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_2971_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_gm(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_2972_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit0,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(4)
tff(fact_2973_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one,aa_num_num(bit0,N)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_2974_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_2975_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one,aa_num_num(bit1,N)) = none_num ).

% and_not_num.simps(3)
tff(fact_2976_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(7)
tff(fact_2977_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( bit_and_not_num(M,N) = some_num(Q2) )
    <=> ( bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_2978_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_gn(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_2979_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none_num )
    <=> ( bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_2980_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),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_2981_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : bit_se725231765392027082nd_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_2982_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,fract(A,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),times_times_int(A,B))) ) ).

% positive_rat
tff(fact_2983_Rat_Opositive__add,axiom,
    ! [X: rat,Y2: rat] :
      ( pp(aa_rat_bool(positive,X))
     => ( pp(aa_rat_bool(positive,Y2))
       => pp(aa_rat_bool(positive,plus_plus_rat(X,Y2))) ) ) ).

% Rat.positive_add
tff(fact_2984_Rat_Opositive__mult,axiom,
    ! [X: rat,Y2: rat] :
      ( pp(aa_rat_bool(positive,X))
     => ( pp(aa_rat_bool(positive,Y2))
       => pp(aa_rat_bool(positive,times_times_rat(X,Y2))) ) ) ).

% Rat.positive_mult
tff(fact_2985_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),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_2986_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y2: option_num] :
      ( ( bit_and_not_num(X,Xa2) = Y2 )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y2 != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N3: num] : Xa2 = aa_num_num(bit0,N3)
             => ( Y2 != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N3: num] : Xa2 = aa_num_num(bit1,N3)
               => ( Y2 != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y2 != some_num(aa_num_num(bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N3: num] :
                        ( ( Xa2 = aa_num_num(bit0,N3) )
                       => ( Y2 != map_option_num_num(bit0,bit_and_not_num(M5,N3)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N3: num] :
                          ( ( Xa2 = aa_num_num(bit1,N3) )
                         => ( Y2 != map_option_num_num(bit0,bit_and_not_num(M5,N3)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y2 != some_num(aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N3: num] :
                              ( ( Xa2 = aa_num_num(bit0,N3) )
                             => ( Y2 != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_gl,bit_and_not_num(M5,N3)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N3: num] :
                                ( ( Xa2 = aa_num_num(bit1,N3) )
                               => ( Y2 != map_option_num_num(bit0,bit_and_not_num(M5,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_2987_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit0,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(5)
tff(fact_2988_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(6)
tff(fact_2989_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(9)
tff(fact_2990_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y2: option_num] :
      ( ( bit_and_not_num(X,Xa2) = Y2 )
     => ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y2 = none_num )
               => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa2 = aa_num_num(bit0,N3) )
                 => ( ( Y2 = some_num(one) )
                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa2 = aa_num_num(bit1,N3) )
                   => ( ( Y2 = none_num )
                     => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y2 = some_num(aa_num_num(bit0,M5)) )
                       => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N3: num] :
                          ( ( Xa2 = aa_num_num(bit0,N3) )
                         => ( ( Y2 = map_option_num_num(bit0,bit_and_not_num(M5,N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N3: num] :
                            ( ( Xa2 = aa_num_num(bit1,N3) )
                           => ( ( Y2 = map_option_num_num(bit0,bit_and_not_num(M5,N3)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y2 = some_num(aa_num_num(bit0,M5)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N3: num] :
                                ( ( Xa2 = aa_num_num(bit0,N3) )
                               => ( ( Y2 = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_gl,bit_and_not_num(M5,N3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N3: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N3) )
                                 => ( ( Y2 = map_option_num_num(bit0,bit_and_not_num(M5,N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_2991_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y2: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,X),Xa2) = Y2 )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y2 != some_num(one) ) ) )
       => ( ( ( X = one )
           => ( ? [N3: num] : Xa2 = aa_num_num(bit0,N3)
             => ( Y2 != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N3: num] : Xa2 = aa_num_num(bit1,N3)
               => ( Y2 != some_num(one) ) ) )
           => ( ( ? [M5: num] : X = aa_num_num(bit0,M5)
               => ( ( Xa2 = one )
                 => ( Y2 != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N3: num] :
                        ( ( Xa2 = aa_num_num(bit0,N3) )
                       => ( Y2 != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N3: num] :
                          ( ( Xa2 = aa_num_num(bit1,N3) )
                         => ( Y2 != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) ) ) )
                 => ( ( ? [M5: num] : X = aa_num_num(bit1,M5)
                     => ( ( Xa2 = one )
                       => ( Y2 != some_num(one) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N3: num] :
                              ( ( Xa2 = aa_num_num(bit0,N3) )
                             => ( Y2 != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N3: num] :
                                ( ( Xa2 = aa_num_num(bit1,N3) )
                               => ( Y2 != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_gl,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_2992_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(5)
tff(fact_2993_and__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),one) = some_num(one) ).

% and_num.simps(1)
tff(fact_2994_and__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,N)) = some_num(one) ).

% and_num.simps(3)
tff(fact_2995_and__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_2996_and__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,N)) = none_num ).

% and_num.simps(2)
tff(fact_2997_and__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),one) = none_num ).

% and_num.simps(4)
tff(fact_2998_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(6)
tff(fact_2999_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(8)
tff(fact_3000_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_gl,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(9)
tff(fact_3001_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y2: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,X),Xa2) = Y2 )
     => ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y2 = some_num(one) )
               => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa2 = aa_num_num(bit0,N3) )
                 => ( ( Y2 = none_num )
                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa2 = aa_num_num(bit1,N3) )
                   => ( ( Y2 = some_num(one) )
                     => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y2 = none_num )
                       => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N3: num] :
                          ( ( Xa2 = aa_num_num(bit0,N3) )
                         => ( ( Y2 = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N3: num] :
                            ( ( Xa2 = aa_num_num(bit1,N3) )
                           => ( ( Y2 = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y2 = some_num(one) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N3: num] :
                                ( ( Xa2 = aa_num_num(bit0,N3) )
                               => ( ( Y2 = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N3: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N3) )
                                 => ( ( Y2 = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_gl,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M5),N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_3002_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y2: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,X),Xa2) = Y2 )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y2 != none_num ) ) )
       => ( ( ( X = one )
           => ! [N3: num] :
                ( ( Xa2 = aa_num_num(bit0,N3) )
               => ( Y2 != some_num(aa_num_num(bit1,N3)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa2 = aa_num_num(bit1,N3) )
                 => ( Y2 != some_num(aa_num_num(bit0,N3)) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y2 != some_num(aa_num_num(bit1,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N3: num] :
                        ( ( Xa2 = aa_num_num(bit0,N3) )
                       => ( Y2 != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N3: num] :
                          ( ( Xa2 = aa_num_num(bit1,N3) )
                         => ( Y2 != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y2 != some_num(aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N3: num] :
                              ( ( Xa2 = aa_num_num(bit0,N3) )
                             => ( Y2 != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3))) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N3: num] :
                                ( ( Xa2 = aa_num_num(bit1,N3) )
                               => ( Y2 != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_3003_xor__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),one) = none_num ).

% xor_num.simps(1)
tff(fact_3004_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(5)
tff(fact_3005_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(9)
tff(fact_3006_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),one) = some_num(aa_num_num(bit0,M)) ).

% xor_num.simps(7)
tff(fact_3007_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),one) = some_num(aa_num_num(bit1,M)) ).

% xor_num.simps(4)
tff(fact_3008_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,N)) = some_num(aa_num_num(bit0,N)) ).

% xor_num.simps(3)
tff(fact_3009_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,N)) = some_num(aa_num_num(bit1,N)) ).

% xor_num.simps(2)
tff(fact_3010_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(6)
tff(fact_3011_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(8)
tff(fact_3012_xor__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y2: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,X),Xa2) = Y2 )
     => ( accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(X,Xa2))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y2 = none_num )
               => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N3: num] :
                  ( ( Xa2 = aa_num_num(bit0,N3) )
                 => ( ( Y2 = some_num(aa_num_num(bit1,N3)) )
                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N3))) ) ) )
           => ( ( ( X = one )
               => ! [N3: num] :
                    ( ( Xa2 = aa_num_num(bit1,N3) )
                   => ( ( Y2 = some_num(aa_num_num(bit0,N3)) )
                     => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N3))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y2 = some_num(aa_num_num(bit1,M5)) )
                       => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N3: num] :
                          ( ( Xa2 = aa_num_num(bit0,N3) )
                         => ( ( Y2 = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3)) )
                           => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N3))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N3: num] :
                            ( ( Xa2 = aa_num_num(bit1,N3) )
                           => ( ( Y2 = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3))) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N3))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y2 = some_num(aa_num_num(bit0,M5)) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N3: num] :
                                ( ( Xa2 = aa_num_num(bit0,N3) )
                               => ( ( Y2 = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3))) )
                                 => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N3))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N3: num] :
                                  ( ( Xa2 = aa_num_num(bit1,N3) )
                                 => ( ( Y2 = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M5),N3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_3013_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_3014_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_3015_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_3016_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_3017_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_gr,product_Pair_nat_num(N,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_3018_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(ord_le3102999989581377725nteger(K,one_one_Code_integer))
       => ( code_num_of_integer(K) = one ) )
      & ( ~ pp(ord_le3102999989581377725nteger(K,one_one_Code_integer))
       => ( code_num_of_integer(K) = produc7336495610019696514er_num(aTP_Lamm_gs,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_3019_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N)))
       => ( upt(numeral_numeral_nat(M),numeral_numeral_nat(N)) = cons_nat(numeral_numeral_nat(M),upt(aa_nat_nat(suc,numeral_numeral_nat(M)),numeral_numeral_nat(N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(N)))
       => ( upt(numeral_numeral_nat(M),numeral_numeral_nat(N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_3020_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups_nat(upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_3021_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( hd_nat(upt(I2,J)) = I2 ) ) ).

% hd_upt
tff(fact_3022_drop__upt,axiom,
    ! [M: nat,I2: nat,J: nat] : drop_nat(M,upt(I2,J)) = upt(aa_nat_nat(plus_plus_nat(I2),M),J) ).

% drop_upt
tff(fact_3023_length__upt,axiom,
    ! [I2: nat,J: nat] : size_size_list_nat(upt(I2,J)) = aa_nat_nat(minus_minus_nat(J),I2) ).

% length_upt
tff(fact_3024_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(I2),M)),N))
     => ( take_nat(M,upt(I2,N)) = upt(I2,aa_nat_nat(plus_plus_nat(I2),M)) ) ) ).

% take_upt
tff(fact_3025_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( upt(I2,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_3026_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(M,N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_3027_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( upt(I2,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_3028_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K)),J))
     => ( aa_nat_nat(nth_nat(upt(I2,J)),K) = aa_nat_nat(plus_plus_nat(I2),K) ) ) ).

% nth_upt
tff(fact_3029_atMost__upto,axiom,
    ! [N: nat] : set_ord_atMost_nat(N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_3030_atLeastLessThan__upt,axiom,
    ! [I2: nat,J: nat] : set_or4665077453230672383an_nat(I2,J) = set_nat2(upt(I2,J)) ).

% atLeastLessThan_upt
tff(fact_3031_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1269000886237332187st_nat(N,M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_3032_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5834768355832116004an_nat(N,M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_3033_atLeast__upt,axiom,
    ! [N: nat] : set_ord_lessThan_nat(N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_3034_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or6659071591806873216st_nat(N,M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_3035_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q2) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_3036_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_3037_distinct__upt,axiom,
    ! [I2: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I2,J))) ).

% distinct_upt
tff(fact_3038_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( upt(I2,J) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J)) ) ) ).

% upt_conv_Cons
tff(fact_3039_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( upt(I2,aa_nat_nat(plus_plus_nat(J),K)) = append_nat(upt(I2,J),upt(J,aa_nat_nat(plus_plus_nat(J),K))) ) ) ).

% upt_add_eq_append
tff(fact_3040_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I2,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
        & ( I2 = X )
        & ( upt(aa_nat_nat(plus_plus_nat(I2),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_3041_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
       => ( upt(I2,J) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
       => ( upt(I2,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_3042_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( upt(I2,aa_nat_nat(suc,J)) = append_nat(upt(I2,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_3043_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
       => ( upt(I2,aa_nat_nat(suc,J)) = append_nat(upt(I2,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
       => ( upt(I2,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_3044_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_al),set_or4665077453230672383an_nat(M,N)) ) ) ).

% sum_list_upt
tff(fact_3045_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_3046_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aTP_Lamm_gt(N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(plus_plus_nat(M),N)) ).

% map_add_upt
tff(fact_3047_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_gu,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_3048_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_gv,Qr) ).

% Divides.adjust_div_def
tff(fact_3049_card__length__sum__list__rec,axiom,
    ! [M: nat,N2: 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_gw(M),N2))) = aa_nat_nat(plus_plus_nat(finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_gx(M),N2)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_gy(M),N2)))) ) ) ).

% card_length_sum_list_rec
tff(fact_3050_card__length__sum__list,axiom,
    ! [M: nat,N2: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_gw(M),N2))) = aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(N2),M)),one_one_nat)),N2) ).

% card_length_sum_list
tff(fact_3051_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_3052_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_3053_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: 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,I2),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_3054_sorted__upto,axiom,
    ! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).

% sorted_upto
tff(fact_3055_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I2,J))) ).

% sorted_wrt_upto
tff(fact_3056_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_3057_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(plus_plus_nat(aa_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_3058_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(plus_plus_nat(aa_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_3059_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_gz(M))) = produc457027306803732586at_nat(set_ord_atMost_nat(M),aTP_Lamm_ha(M)) ).

% pairs_le_eq_Sigma
tff(fact_3060_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( finite_finite_nat(S3)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(S3)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S3)))) ) ).

% card_le_Suc_Max
tff(fact_3061_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(divide_divide_nat(M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(divide_divide_nat(M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hb(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_3062_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(gcd_gcd_nat(M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hc(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_3063_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,M)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_3064_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_3065_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_3066_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_hd) ).

% prod_encode_def
tff(fact_3067_list__encode_Oelims,axiom,
    ! [X: list_nat,Y2: nat] :
      ( ( nat_list_encode(X) = Y2 )
     => ( ( ( X = nil_nat )
         => ( Y2 != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X3,Xs2) )
             => ( Y2 != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_3068_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_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),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_3069_list__encode_Opelims,axiom,
    ! [X: list_nat,Y2: nat] :
      ( ( nat_list_encode(X) = Y2 )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y2 = 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) )
               => ( ( Y2 = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),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_3070_Gcd__nat__eq__one,axiom,
    ! [N2: set_nat] :
      ( member_nat(one_one_nat,N2)
     => ( gcd_Gcd_nat(N2) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_3071_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_3072_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_3073_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_he) ).

% Rat.positive_def
tff(fact_3074_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_al,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_3075_sort__upto,axiom,
    ! [I2: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_ag,upto(I2,J)) = upto(I2,J) ).

% sort_upto
tff(fact_3076_ATP_Olambda__1,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_ax,Uu2) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu2)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_3077_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_as,Uu2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_2
tff(fact_3078_ATP_Olambda__3,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_bz,Uu2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu2),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_3079_ATP_Olambda__4,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_at,Uu2) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu2)) ).

% ATP.lambda_4
tff(fact_3080_ATP_Olambda__5,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ew,Uu2) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).

% ATP.lambda_5
tff(fact_3081_ATP_Olambda__6,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_ez,Uu2) = aa_real_real(aa_nat_fun_real_real(root,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_6
tff(fact_3082_ATP_Olambda__7,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_gu,Uu2) = aa_nat_nat(minus_minus_nat(Uu2),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_7
tff(fact_3083_ATP_Olambda__8,axiom,
    ! [Uu2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_he,Uu2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),times_times_int(product_fst_int_int(Uu2),product_snd_int_int(Uu2)))) ) ).

% ATP.lambda_8
tff(fact_3084_ATP_Olambda__9,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_fc,Uu2) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_9
tff(fact_3085_ATP_Olambda__10,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_eu,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_aw,Uu2)) ).

% ATP.lambda_10
tff(fact_3086_ATP_Olambda__11,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_fe,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).

% ATP.lambda_11
tff(fact_3087_ATP_Olambda__12,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_gl,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).

% ATP.lambda_12
tff(fact_3088_ATP_Olambda__13,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_gk,Uu2) = some_num(aa_num_num(bit0,Uu2)) ).

% ATP.lambda_13
tff(fact_3089_ATP_Olambda__14,axiom,
    ! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_aj,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_14
tff(fact_3090_ATP_Olambda__15,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_gg,Uu2))
    <=> ? [I4: int,N4: nat] :
          ( ( Uu2 = aa_real_real(divide_divide_real(ring_1_of_int_real(I4)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_15
tff(fact_3091_ATP_Olambda__16,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_gh,Uu2))
    <=> ? [I4: int,J3: int] :
          ( ( Uu2 = aa_real_real(divide_divide_real(ring_1_of_int_real(I4)),ring_1_of_int_real(J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_16
tff(fact_3092_ATP_Olambda__17,axiom,
    ! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_gj,Uu2) = some_num(one) ).

% ATP.lambda_17
tff(fact_3093_ATP_Olambda__18,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_gq(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_go(Uua),aTP_Lamm_gp(Uua),Uu2) ).

% ATP.lambda_18
tff(fact_3094_ATP_Olambda__19,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_an(Uu2),Uua) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua),zero_zero_real),aa_nat_real(Uu2,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Uua),one_one_nat)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% ATP.lambda_19
tff(fact_3095_ATP_Olambda__20,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_dn,Uu2),Uua) = if_int(fequal_Code_integer(Uua,zero_z3403309356797280102nteger),times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),code_int_of_integer(Uu2)),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),code_int_of_integer(Uu2)),one_one_int)) ).

% ATP.lambda_20
tff(fact_3096_ATP_Olambda__21,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_gs,Uu2),Uua) = if_num(fequal_Code_integer(Uua,zero_z3403309356797280102nteger),plus_plus_num(code_num_of_integer(Uu2),code_num_of_integer(Uu2)),plus_plus_num(plus_plus_num(code_num_of_integer(Uu2),code_num_of_integer(Uu2)),one)) ).

% ATP.lambda_21
tff(fact_3097_ATP_Olambda__22,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_dl,Uu2),Uua) = aa_nat_nat(if_nat(fequal_Code_integer(Uua,zero_z3403309356797280102nteger),aa_nat_nat(plus_plus_nat(code_nat_of_integer(Uu2)),code_nat_of_integer(Uu2))),aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(code_nat_of_integer(Uu2)),code_nat_of_integer(Uu2))),one_one_nat)) ).

% ATP.lambda_22
tff(fact_3098_ATP_Olambda__23,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_cy,Uu2),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu2),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(times_times_int(sgn_sgn_int(Uu2),Uua),aa_int_int(abs_abs_int,Uu2))) ).

% ATP.lambda_23
tff(fact_3099_ATP_Olambda__24,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_gr,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_gq(Uua),Uu2) ).

% ATP.lambda_24
tff(fact_3100_ATP_Olambda__25,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_go(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_gk,bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_25
tff(fact_3101_ATP_Olambda__26,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_gm(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_gk,bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_26
tff(fact_3102_ATP_Olambda__27,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_au,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(plus_plus_nat(Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_27
tff(fact_3103_ATP_Olambda__28,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_be,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_28
tff(fact_3104_ATP_Olambda__29,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bg,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_29
tff(fact_3105_ATP_Olambda__30,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_bw,Uu2),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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
        & ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_30
tff(fact_3106_ATP_Olambda__31,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_bv,Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),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),numeral_numeral_real(aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_31
tff(fact_3107_ATP_Olambda__32,axiom,
    ! [Uu2: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ca(Uu2),Uua))
    <=> ( ( sgn_sgn_complex(Uu2) = 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_32
tff(fact_3108_ATP_Olambda__33,axiom,
    ! [Uu2: real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_cw(Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,ring_1_of_int_real(Uua)),Uu2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu2),ring_1_of_int_real(plus_plus_int(Uua,one_one_int)))) ) ) ).

% ATP.lambda_33
tff(fact_3109_ATP_Olambda__34,axiom,
    ! [Uu2: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_cx(Uu2),Uua))
    <=> ( ord_less_eq_rat(ring_1_of_int_rat(Uua),Uu2)
        & ord_less_rat(Uu2,ring_1_of_int_rat(plus_plus_int(Uua,one_one_int))) ) ) ).

% ATP.lambda_34
tff(fact_3110_ATP_Olambda__35,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aw,Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_35
tff(fact_3111_ATP_Olambda__36,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ev,Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% ATP.lambda_36
tff(fact_3112_ATP_Olambda__37,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ex(Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_37
tff(fact_3113_ATP_Olambda__38,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_bs(Uu2),Uua) = aa_nat_nat(power_power_nat(aa_nat_nat(binomial(Uu2),Uua)),numeral_numeral_nat(aa_num_num(bit0,one))) ).

% ATP.lambda_38
tff(fact_3114_ATP_Olambda__39,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_gd(Uu2),Uua))
    <=> ( member_int(Uua,Uu2)
        & ! [X4: int] :
            ( member_int(X4,Uu2)
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Uua)) ) ) ) ).

% ATP.lambda_39
tff(fact_3115_ATP_Olambda__40,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bi,Uu2),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_40
tff(fact_3116_ATP_Olambda__41,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_fs,Uu2),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_41
tff(fact_3117_ATP_Olambda__42,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_fz(Uu2),Uua) = aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(Uu2),Uua)),Uua) ).

% ATP.lambda_42
tff(fact_3118_ATP_Olambda__43,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_bl(Uu2),Uua) = aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(Uu2),Uua)),Uu2) ).

% ATP.lambda_43
tff(fact_3119_ATP_Olambda__44,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_bk(Uu2),Uua) = aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(Uu2),Uua)),Uua) ).

% ATP.lambda_44
tff(fact_3120_ATP_Olambda__45,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_bc(Uu2),Uua))
    <=> ( power_power_complex(Uua,Uu2) = one_one_complex ) ) ).

% ATP.lambda_45
tff(fact_3121_ATP_Olambda__46,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fj,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_46
tff(fact_3122_ATP_Olambda__47,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aTP_Lamm_fq(Uu2),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(Uu2),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_47
tff(fact_3123_ATP_Olambda__48,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aTP_Lamm_ft(Uu2),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu2),Uua))),Uua) ).

% ATP.lambda_48
tff(fact_3124_ATP_Olambda__49,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_av,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_49
tff(fact_3125_ATP_Olambda__50,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_by,Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_50
tff(fact_3126_ATP_Olambda__51,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fp(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_ex(Uu2),set_ord_lessThan_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_51
tff(fact_3127_ATP_Olambda__52,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fo(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_ex(Uu2),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_52
tff(fact_3128_ATP_Olambda__53,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hd,Uu2),Uua) = aa_nat_nat(plus_plus_nat(aa_nat_nat(nat_triangle,aa_nat_nat(plus_plus_nat(Uu2),Uua))),Uu2) ).

% ATP.lambda_53
tff(fact_3129_ATP_Olambda__54,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_gi(Uu2),Uua) = bit_se2923211474154528505it_int(numeral_numeral_nat(Uu2),minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),numeral_numeral_nat(Uu2)),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_54
tff(fact_3130_ATP_Olambda__55,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ec,Uu2),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu2)) ).

% ATP.lambda_55
tff(fact_3131_ATP_Olambda__56,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bh,Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_56
tff(fact_3132_ATP_Olambda__57,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bd,Uu2),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_57
tff(fact_3133_ATP_Olambda__58,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_bt(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(binomial(Uu2),Uua)) ).

% ATP.lambda_58
tff(fact_3134_ATP_Olambda__59,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fk,Uu2),Uua) = aa_real_real(times_times_real(Uu2),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_59
tff(fact_3135_ATP_Olambda__60,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_gv,Uu2),Uua) = plus_plus_int(Uu2,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_60
tff(fact_3136_ATP_Olambda__61,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fi,Uu2),Uua) = aa_real_real(plus_plus_real(Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_61
tff(fact_3137_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ff,Uu2),Uua) = aa_real_real(plus_plus_real(Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_62
tff(fact_3138_ATP_Olambda__63,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_dz(Uu2),Uua) = groups6591440286371151544t_real(Uu2,set_ord_lessThan_nat(Uua)) ).

% ATP.lambda_63
tff(fact_3139_ATP_Olambda__64,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gb,Uu2),Uua))
    <=> ( Uua = aa_nat_nat(suc,Uu2) ) ) ).

% ATP.lambda_64
tff(fact_3140_ATP_Olambda__65,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cg,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu2)) ) ).

% ATP.lambda_65
tff(fact_3141_ATP_Olambda__66,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_eb(Uu2),Uua) = aa_nat_nat(modulo_modulo_nat(Uua),Uu2) ).

% ATP.lambda_66
tff(fact_3142_ATP_Olambda__67,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ak,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu2)) ) ).

% ATP.lambda_67
tff(fact_3143_ATP_Olambda__68,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ey(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ).

% ATP.lambda_68
tff(fact_3144_ATP_Olambda__69,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_dy(Uu2),Uua) = aa_nat_nat(minus_minus_nat(Uua),Uu2) ).

% ATP.lambda_69
tff(fact_3145_ATP_Olambda__70,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ed,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ).

% ATP.lambda_70
tff(fact_3146_ATP_Olambda__71,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gt(Uu2),Uua) = aa_nat_nat(plus_plus_nat(Uua),Uu2) ).

% ATP.lambda_71
tff(fact_3147_ATP_Olambda__72,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_ea(Uu2),Uua) = plus_plus_int(Uua,Uu2) ).

% ATP.lambda_72
tff(fact_3148_ATP_Olambda__73,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aTP_Lamm_ef(Uu2),Uua) = aa_real_real(powr_real(Uua),Uu2) ).

% ATP.lambda_73
tff(fact_3149_ATP_Olambda__74,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cf,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu2)) ) ).

% ATP.lambda_74
tff(fact_3150_ATP_Olambda__75,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_bj(Uu2),Uua) = aa_nat_nat(binomial(Uua),Uu2) ).

% ATP.lambda_75
tff(fact_3151_ATP_Olambda__76,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fd,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ).

% ATP.lambda_76
tff(fact_3152_ATP_Olambda__77,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_a(Uu2),Uua))
    <=> member8440522571783428010at_nat(Uua,Uu2) ) ).

% ATP.lambda_77
tff(fact_3153_ATP_Olambda__78,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_aa(Uu2),Uua))
    <=> member_complex(Uua,Uu2) ) ).

% ATP.lambda_78
tff(fact_3154_ATP_Olambda__79,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ab(Uu2),Uua))
    <=> member_real(Uua,Uu2) ) ).

% ATP.lambda_79
tff(fact_3155_ATP_Olambda__80,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ac(Uu2),Uua))
    <=> member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_80
tff(fact_3156_ATP_Olambda__81,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu2),Uua))
    <=> member_nat(Uua,Uu2) ) ).

% ATP.lambda_81
tff(fact_3157_ATP_Olambda__82,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu2),Uua))
    <=> member_int(Uua,Uu2) ) ).

% ATP.lambda_82
tff(fact_3158_ATP_Olambda__83,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ar(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_83
tff(fact_3159_ATP_Olambda__84,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aq(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_84
tff(fact_3160_ATP_Olambda__85,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_fu(Uu2),Uua))
    <=> pp(aa_nat_bool(Uu2,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_85
tff(fact_3161_ATP_Olambda__86,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_gp(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_86
tff(fact_3162_ATP_Olambda__87,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_gn(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_87
tff(fact_3163_ATP_Olambda__88,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_dx,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_dw(Uu2),Uua)) ).

% ATP.lambda_88
tff(fact_3164_ATP_Olambda__89,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_dv,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_du(Uu2),Uua)) ).

% ATP.lambda_89
tff(fact_3165_ATP_Olambda__90,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_dt,Uu2),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ds(Uu2),Uua)) ).

% ATP.lambda_90
tff(fact_3166_ATP_Olambda__91,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_dr,Uu2),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_dq(Uu2),Uua)) ).

% ATP.lambda_91
tff(fact_3167_ATP_Olambda__92,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_dp,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_do(Uu2),Uua)) ).

% ATP.lambda_92
tff(fact_3168_ATP_Olambda__93,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_em(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_el(Uu2),Uua)) ).

% ATP.lambda_93
tff(fact_3169_ATP_Olambda__94,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_bu(Uu2),Uua) = cis(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).

% ATP.lambda_94
tff(fact_3170_ATP_Olambda__95,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_af,Uu2),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(divide_divide_nat(Uu2),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_95
tff(fact_3171_ATP_Olambda__96,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_ha(Uu2),Uua) = set_ord_atMost_nat(aa_nat_nat(minus_minus_nat(Uu2),Uua)) ).

% ATP.lambda_96
tff(fact_3172_ATP_Olambda__97,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fh,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_97
tff(fact_3173_ATP_Olambda__98,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gf(Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu2),Uua)) ).

% ATP.lambda_98
tff(fact_3174_ATP_Olambda__99,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ge(Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu2)) ).

% ATP.lambda_99
tff(fact_3175_ATP_Olambda__100,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ct(Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu2),Uua)) ).

% ATP.lambda_100
tff(fact_3176_ATP_Olambda__101,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cs(Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu2)) ).

% ATP.lambda_101
tff(fact_3177_ATP_Olambda__102,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fb(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_102
tff(fact_3178_ATP_Olambda__103,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_fy(Uu2),Uua) = aa_real_real(artanh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_103
tff(fact_3179_ATP_Olambda__104,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_fx(Uu2),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_104
tff(fact_3180_ATP_Olambda__105,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ay(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_105
tff(fact_3181_ATP_Olambda__106,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fn(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_106
tff(fact_3182_ATP_Olambda__107,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_ej(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ).

% ATP.lambda_107
tff(fact_3183_ATP_Olambda__108,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ao(Uu2),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(divide_divide_nat(Uub),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_real(Uu2,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Uub),one_one_nat)),numeral_numeral_nat(aa_num_num(bit0,one))))) ).

% ATP.lambda_108
tff(fact_3184_ATP_Olambda__109,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ap(Uu2),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uub),aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_109
tff(fact_3185_ATP_Olambda__110,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_cm(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(ord_le3102999989581377725nteger(aa_num_Code_integer(numera6620942414471956472nteger,Uu2),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,aa_num_Code_integer(numera6620942414471956472nteger,Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_110
tff(fact_3186_ATP_Olambda__111,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ah(Uu2),Uua),Uub) = if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(Uu2)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(minus_minus_nat(Uub),numeral_numeral_nat(Uu2))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_111
tff(fact_3187_ATP_Olambda__112,axiom,
    ! [Uu2: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ai(Uu2),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),Uub),product_Pair_int_int(plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Uua),one_one_int),minus_minus_int(Uub,aa_num_int(numeral_numeral_int,Uu2))),product_Pair_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Uua),Uub)) ).

% ATP.lambda_112
tff(fact_3188_ATP_Olambda__113,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_cv(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(fequal_Code_integer(Uub,zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu2),Uub))) ).

% ATP.lambda_113
tff(fact_3189_ATP_Olambda__114,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dm(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(fequal_Code_integer(Uub,zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu2),Uub))) ).

% ATP.lambda_114
tff(fact_3190_ATP_Olambda__115,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_cu(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(fequal_Code_integer(Uub,zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu2,Uub))) ).

% ATP.lambda_115
tff(fact_3191_ATP_Olambda__116,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ei(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_116
tff(fact_3192_ATP_Olambda__117,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_cp(Uu2),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,Uu2),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),fequal_Code_integer(Uub,one_one_Code_integer)) ).

% ATP.lambda_117
tff(fact_3193_ATP_Olambda__118,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dk(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dj(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_118
tff(fact_3194_ATP_Olambda__119,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_di(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dh(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_119
tff(fact_3195_ATP_Olambda__120,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dg(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_df(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_120
tff(fact_3196_ATP_Olambda__121,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_de(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dd(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_121
tff(fact_3197_ATP_Olambda__122,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dc(Uu2),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_db(Uua),Uub)),quotient_of(Uu2))) ) ).

% ATP.lambda_122
tff(fact_3198_ATP_Olambda__123,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_da(Uu2),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_cz(Uua),Uub)),quotient_of(Uu2))) ) ).

% ATP.lambda_123
tff(fact_3199_ATP_Olambda__124,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_en(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_124
tff(fact_3200_ATP_Olambda__125,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_eo(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_125
tff(fact_3201_ATP_Olambda__126,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ek(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_126
tff(fact_3202_ATP_Olambda__127,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_bf(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_127
tff(fact_3203_ATP_Olambda__128,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_gx(Uu2),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(minus_minus_nat(Uu2),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_128
tff(fact_3204_ATP_Olambda__129,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_gy(Uu2),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( aa_nat_nat(plus_plus_nat(groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_129
tff(fact_3205_ATP_Olambda__130,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_gw(Uu2),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_130
tff(fact_3206_ATP_Olambda__131,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cn(Uu2),Uua),Uub))
    <=> ( member_nat(aa_nat_nat(suc,Uub),Uu2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_131
tff(fact_3207_ATP_Olambda__132,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_co(Uu2),Uua),Uub))
    <=> ( member_nat(Uub,Uu2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_132
tff(fact_3208_ATP_Olambda__133,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bm(Uu2),Uua),Uub) = aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Uua),Uub)),aa_nat_nat(minus_minus_nat(Uu2),Uub)) ).

% ATP.lambda_133
tff(fact_3209_ATP_Olambda__134,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cc(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_134
tff(fact_3210_ATP_Olambda__135,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ch(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_135
tff(fact_3211_ATP_Olambda__136,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cb(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_136
tff(fact_3212_ATP_Olambda__137,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ci(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_137
tff(fact_3213_ATP_Olambda__138,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cj(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_138
tff(fact_3214_ATP_Olambda__139,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_hc(Uu2),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),Uu2)) ) ) ).

% ATP.lambda_139
tff(fact_3215_ATP_Olambda__140,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_hb(Uu2),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),Uu2)),Uua)) ) ).

% ATP.lambda_140
tff(fact_3216_ATP_Olambda__141,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gz(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(Uua),Uub)),Uu2)) ) ).

% ATP.lambda_141
tff(fact_3217_ATP_Olambda__142,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_bb(Uu2),Uua),Uub))
    <=> ( power_power_complex(Uub,Uu2) = Uua ) ) ).

% ATP.lambda_142
tff(fact_3218_ATP_Olambda__143,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_ck(Uu2),Uua),Uub))
    <=> ( power_power_complex(Uub,Uua) = Uu2 ) ) ).

% ATP.lambda_143
tff(fact_3219_ATP_Olambda__144,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_fm(Uu2),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),aa_real_real(times_times_real(ring_1_of_int_real(aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_144
tff(fact_3220_ATP_Olambda__145,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_el(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_145
tff(fact_3221_ATP_Olambda__146,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_az(Uu2),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_146
tff(fact_3222_ATP_Olambda__147,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bp(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(power_power_nat(Uua),Uub)) ).

% ATP.lambda_147
tff(fact_3223_ATP_Olambda__148,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cd(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu2,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_148
tff(fact_3224_ATP_Olambda__149,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cl(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_149
tff(fact_3225_ATP_Olambda__150,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(Uu2),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_150
tff(fact_3226_ATP_Olambda__151,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_eh(Uu2),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_151
tff(fact_3227_ATP_Olambda__152,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ce(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua)) ) ).

% ATP.lambda_152
tff(fact_3228_ATP_Olambda__153,axiom,
    ! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ee(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_153
tff(fact_3229_ATP_Olambda__154,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fr(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ).

% ATP.lambda_154
tff(fact_3230_ATP_Olambda__155,axiom,
    ! [Uu2: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_eg(Uu2),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_155
tff(fact_3231_ATP_Olambda__156,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fg(Uu2),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_156
tff(fact_3232_ATP_Olambda__157,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_am(Uu2),Uua),Uub) = aa_nat_nat(plus_plus_nat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_157
tff(fact_3233_ATP_Olambda__158,axiom,
    ! [Uu2: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_fw(Uu2),Uua),Uub))
    <=> pp(aa_real_bool(Uu2,aa_real_real(plus_plus_real(Uub),Uua))) ) ).

% ATP.lambda_158
tff(fact_3234_ATP_Olambda__159,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fv(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(Uu2,aa_nat_nat(plus_plus_nat(Uub),Uua))) ) ).

% ATP.lambda_159
tff(fact_3235_ATP_Olambda__160,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_ga(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_160
tff(fact_3236_ATP_Olambda__161,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fl(Uu2),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),Uua)) ).

% ATP.lambda_161
tff(fact_3237_ATP_Olambda__162,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_br(Uu2),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_bq(Uu2),Uua),Uuc)),set_ord_atMost_nat(Uuc))),aa_nat_nat(power_power_nat(Uub),Uuc)) ).

% ATP.lambda_162
tff(fact_3238_ATP_Olambda__163,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_et(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(plus_plus_nat(aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_163
tff(fact_3239_ATP_Olambda__164,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_er(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(plus_plus_nat(Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_164
tff(fact_3240_ATP_Olambda__165,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ep(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_165
tff(fact_3241_ATP_Olambda__166,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_eq(Uu2),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_166
tff(fact_3242_ATP_Olambda__167,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_do(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_167
tff(fact_3243_ATP_Olambda__168,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_bo(Uu2),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(binomial(Uub),Uuc))),aa_nat_nat(power_power_nat(Uu2),Uuc))),aa_nat_nat(power_power_nat(Uua),aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_168
tff(fact_3244_ATP_Olambda__169,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_bn(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Uu2),Uuc)),aa_nat_nat(binomial(Uua),aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_169
tff(fact_3245_ATP_Olambda__170,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_db(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,times_times_int(Uu2,Uuc)),times_times_int(Uua,Uub))) ) ).

% ATP.lambda_170
tff(fact_3246_ATP_Olambda__171,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ds(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(Uu2),Uuc)),aa_nat_nat(plus_plus_nat(Uub),Uua))) ) ).

% ATP.lambda_171
tff(fact_3247_ATP_Olambda__172,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_cz(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,times_times_int(Uu2,Uuc)),times_times_int(Uua,Uub))) ) ).

% ATP.lambda_172
tff(fact_3248_ATP_Olambda__173,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_dq(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(plus_plus_nat(Uu2),Uuc)),aa_nat_nat(plus_plus_nat(Uub),Uua))) ) ).

% ATP.lambda_173
tff(fact_3249_ATP_Olambda__174,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_du(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(plus_plus_nat(Uu2),Uub)),aa_nat_nat(plus_plus_nat(Uua),Uuc)) ).

% ATP.lambda_174
tff(fact_3250_ATP_Olambda__175,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_dw(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(plus_plus_nat(Uu2),Uuc)),aa_nat_nat(plus_plus_nat(Uua),Uub)) ).

% ATP.lambda_175
tff(fact_3251_ATP_Olambda__176,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_bq(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_176
tff(fact_3252_ATP_Olambda__177,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_df(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(minus_minus_int(times_times_int(Uu2,Uuc),times_times_int(Uub,Uua)),times_times_int(Uua,Uuc))) ).

% ATP.lambda_177
tff(fact_3253_ATP_Olambda__178,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dd(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(plus_plus_int(times_times_int(Uu2,Uuc),times_times_int(Uub,Uua)),times_times_int(Uua,Uuc))) ).

% ATP.lambda_178
tff(fact_3254_ATP_Olambda__179,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dh(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(times_times_int(Uu2,Uub),times_times_int(Uua,Uuc))) ).

% ATP.lambda_179
tff(fact_3255_ATP_Olambda__180,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dj(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(times_times_int(Uu2,Uuc),times_times_int(Uua,Uub))) ).

% ATP.lambda_180
tff(fact_3256_ATP_Olambda__181,axiom,
    ! [Uu2: 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_es(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_er(Uua),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(minus_minus_nat(Uu2),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(minus_minus_nat(Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(minus_minus_nat(Uu2),Uuc)))))) ).

% ATP.lambda_181
tff(fact_3257_ATP_Olambda__182,axiom,
    ! [Uu2: 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_gc(Uu2),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(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Uu2)))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(minus_minus_nat(Uu2),aa_nat_nat(divide_divide_nat(Uu2),numeral_numeral_nat(aa_num_num(bit0,one)))))))
           => ( ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_1))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I4)) ) )
        & ( ( Uuc = Uud )
         => ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(divide_divide_nat(Uu2),numeral_numeral_nat(aa_num_num(bit0,one))),Uua),Uud))
            & ! [X4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),Uu2)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(divide_divide_nat(Uu2),numeral_numeral_nat(aa_num_num(bit0,one))),Uua),X4))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X4))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_182
tff(fact_3258_ATP_Olambda__183,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_ba,Uu2) = Uu2 ).

% ATP.lambda_183
tff(fact_3259_ATP_Olambda__184,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_al,Uu2) = Uu2 ).

% ATP.lambda_184
tff(fact_3260_ATP_Olambda__185,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_ag,Uu2) = Uu2 ).

% ATP.lambda_185
tff(fact_3261_ATP_Olambda__186,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_bx,Uu2))
    <=> $false ) ).

% ATP.lambda_186
tff(fact_3262_ATP_Olambda__187,axiom,
    ! [Uu2: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_cr,Uu2))
    <=> $false ) ).

% ATP.lambda_187
tff(fact_3263_ATP_Olambda__188,axiom,
    ! [Uu2: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_cq,Uu2))
    <=> $true ) ).

% ATP.lambda_188

% Helper facts (50)
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,Y2: int] : if_int(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y2: int] : if_int(fTrue,X,Y2) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] : aa_nat_nat(if_nat(fFalse,X),Y2) = Y2 ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] : aa_nat_nat(if_nat(fTrue,X),Y2) = X ).

tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y2: num] : if_num(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y2: num] : if_num(fTrue,X,Y2) = 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,Y2: real] : aa_real_real(if_real(fFalse,X),Y2) = Y2 ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y2: real] : aa_real_real(if_real(fTrue,X),Y2) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y2: int] :
      ( ( X != Y2 )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y2)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y2))
      | ( X = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y2)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y2))
      | ( X = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y2: bool] :
      ( ( X != Y2 )
      | pp(aa_bool_bool(fequal_bool(X),Y2)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y2: bool] :
      ( ~ pp(aa_bool_bool(fequal_bool(X),Y2))
      | ( X = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y2: real] :
      ( ( X != Y2 )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y2)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y2))
      | ( X = Y2 ) ) ).

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__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y2: code_integer] : if_Code_integer(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y2: code_integer] : if_Code_integer(fTrue,X,Y2) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y2: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y2: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y2) = X ).

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,Y2: code_integer] :
      ( ( X != Y2 )
      | pp(fequal_Code_integer(X,Y2)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y2: code_integer] :
      ( ~ pp(fequal_Code_integer(X,Y2))
      | ( X = Y2 ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y2: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y2: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y2) = 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,Y2: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X,Y2) = 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,Y2: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y2) = 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,
    ! [I3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(power_power_nat(numeral_numeral_nat(aa_num_num(bit0,one))),m)))
      | ( ( ( aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa)),na) != I3 )
          | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(append_VEBT_VEBT(take_VEBT_VEBT(aa_nat_nat(vEBT_VEBT_high(xa),na),treeList),append_VEBT_VEBT(cons_VEBT_VEBT(vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(vEBT_VEBT_low(xa),na)),nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(aa_nat_nat(vEBT_VEBT_high(xa),na)),one_one_nat),treeList))),I3)),aa_nat_nat(vEBT_VEBT_low(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa)),na))) )
        & ! [Y3: nat] :
            ( ( aa_nat_nat(vEBT_VEBT_high(Y3),na) != I3 )
            | ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(append_VEBT_VEBT(take_VEBT_VEBT(aa_nat_nat(vEBT_VEBT_high(xa),na),treeList),append_VEBT_VEBT(cons_VEBT_VEBT(vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(vEBT_VEBT_high(xa),na)),aa_nat_nat(vEBT_VEBT_low(xa),na)),nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(plus_plus_nat(aa_nat_nat(vEBT_VEBT_high(xa),na)),one_one_nat),treeList))),I3)),aa_nat_nat(vEBT_VEBT_low(Y3),na)))
            | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),Y3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,ma),xa))) ) ) ) ) ).

%------------------------------------------------------------------------------